Application Of Integrals Questions

We provide application of integrals practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on application of integrals skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of application of integrals Questions

Question No Questions Class
1 In an isosceles trapezium, the length of one of the parallel sides, and the lengths of the non-parallel sides are all equal to
( 30 . ) In order to maximize the area of the
trapezium, the smallest angle should be
A ( cdot frac{pi}{6} )
B.
( c cdot frac{pi}{3} )
D.
12
2 (i) Find the area bounded by the curve ( boldsymbol{y}=boldsymbol{x}(mathbf{1}-boldsymbol{x}) ) between the points where
it crosses the x-axis.
(ii) Find the area between the curves
( boldsymbol{y}=boldsymbol{x} ) and ( boldsymbol{y}=boldsymbol{x}^{3} )
12
3 Find the area between the region of
parabola ( boldsymbol{y}^{2}=mathbf{2} boldsymbol{x} ) and circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}= )
8
12
4 Find the area of the region bounded by
the curves
( x^{2}+y^{2}=16 ) and ( x^{2}=6 y )
12
5 The area of the region described by
( A=(x, y): x^{2}+y^{2} leq 1 ) and ( y^{2} leq 1-x )
is:
A ( cdot frac{pi}{2}+frac{4}{3} )
В ( cdot frac{pi}{2}-frac{4}{3} )
c. ( frac{pi}{2}-frac{2}{3} )
D. ( frac{pi}{2}+frac{2}{3} )
12
6 ( operatorname{Given} f(x)=left{begin{array}{l}x, 0 leq x<frac{1}{2} \ frac{1}{2}, x=frac{1}{2} \ 1-x, frac{1}{2}<x leq 1end{array} ) and right.
( g(x)=left(x-frac{1}{2}right)^{2}, x epsilon R, ) Then the area
(in sq.units) of the region bounded by the curves ( y=f(x) ) and ( y=g(x) ) between the lines ( 2 x=1 ) and ( 2 x=sqrt{3} )
is
( ^{A} cdot frac{1}{3}+frac{sqrt{3}}{4} )
B. ( frac{1}{2}-frac{sqrt{3}}{4} )
c. ( frac{1}{2}+frac{sqrt{3}}{4} )
D. ( frac{sqrt{3}}{4}-frac{1}{3} )
12
7 The area bounded by the curves ( y= )
( sqrt{5-x^{2}} ) and ( y=|x-1| ) is
( ^{mathbf{A}} cdotleft(frac{5 pi}{4}-2right) ) square unit
B. ( frac{(5 pi-2)}{4} ) square units
c. ( frac{(5 pi-2)}{2} ) square units
D ( cdotleft(frac{5 pi}{2}-2right) ) square unit
12
8 Find the area of the region.
( left{(x, y): x^{2}+y^{2} leq 8, x^{2} leq 2 yright} )
12
9 6.
bounded by the curves y =lnx, y = ln x,y=| In x
and y=| In pxis
[2002]
(a) 4er units
(b) 6 sq. units
(c) 10 sq. units
(d) none of these
12
10 The total area enclosed by the lines ( y= )
( |x|,|x|=1 ) and ( y=0 ) is:
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot frac{1}{2} )
D. 4
12
11 Show the region of feasible solution
under the following constraints ( 2 x+ ) ( boldsymbol{y} leq mathbf{8}, boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} ) in answer book
12
12 A conic passes through the point (2,4) and is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected
at the point of tangency. Then equation of auxiliary circle of the conic is
A ( cdot x^{2}+y^{2}=16 )
B . ( x^{2}+y^{2}=25 )
c. ( x^{2}+y^{2}=4 sqrt{2} )
D. none of these
12
13 The area bounded by the curve ( y= )
( 2 x^{4}-x^{2}, x ) -axis and the two ordinates
corresponding to the minima of the function is ( frac{8}{a} . ) Find ( a )
12
14 The equation of tangent to ( x^{2}=y^{3} ) at
(1,1) is
12
15 The area bounded by the curves ( x+ )
( mathbf{2}|boldsymbol{y}|=mathbf{1} ) and ( boldsymbol{x}=mathbf{0} ) is?
A ( cdot frac{1}{4} )
в. ( frac{1}{2} )
( c .1 )
D. 2
12
16 The area bounded by the curves y = f(x), the x-axis and the
ordinates x=1 and x=b is (b-1) sin (35+4). Then f(x) is
(a) (x – 1) cos (3x +4)
(1982 – 2 Marks)
(b) sin (3x+4)
(c) sin (3x+4) +3 (x – 1) cos (3x+4)
(d) none of these
12
17 The area of the region bounded by the curve ( y=2 x-x^{2} ) and the line ( y=x ) is
square units.
A ( cdot frac{1}{6} )
в. ( frac{1}{2} )
( c cdot frac{1}{3} )
D. ( frac{7}{6} )
12
18 Area bounded by ( y=2 x-x^{2} &(x- )
1) ( ^{2}+y^{2}=1 ) in first quadrant, is:
A ( cdot frac{pi}{2}-frac{4}{3} )
B ( cdot frac{pi}{2}-frac{2}{3} )
c. ( frac{pi}{2}+frac{4}{3} )
D. ( frac{pi}{2}+frac{2}{3} )
12
19 7.
Find the area bounded by the x-axis, part of the curve
y = 1+
and the ordir
and the ordinates at x = 2 and x = 4. If the
ordinate at x= a divides the area into two equal parts, find a.
(1983 – 3 Marks)
12
20 Identify a possible graph for function ( boldsymbol{f} )
given by
( boldsymbol{f}(boldsymbol{x})=-boldsymbol{2}|boldsymbol{x}| )
A. graph a
B. graph b
c. graph c
D. graph d
12
21 Identify the graph of the exponential
function ( boldsymbol{f} )
( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{(boldsymbol{x}+mathbf{2})} )
12
22 Find the area bounded by line ( y=3 x+ )
2, x-axis and ordinates ( x=-1 ) and
( boldsymbol{x}=mathbf{1} )
12
23 What is the area of the region bounded
by the parabola ( boldsymbol{y}^{2}=mathbf{6}(boldsymbol{x}-mathbf{1}) ) and
( boldsymbol{y}^{2}=mathbf{3} boldsymbol{x} )
A ( cdot frac{sqrt{6}}{3} )
в. ( frac{2 sqrt{6}}{3} )
c. ( frac{4 sqrt{6}}{3} )
D. ( frac{5 sqrt{6}}{3} )
12
24 In the given figure, a square ( boldsymbol{O A B C} ) has
been inscribed in the quadrant OPB ( Q )
If ( boldsymbol{O} boldsymbol{A}=mathbf{2 0} boldsymbol{c m} ) then the area of the
shaded region is ( [text {take } pi=mathbf{3 . 1 4}] )
A ( cdot 214 c m^{2} )
B. ( 228 c m^{2} )
( c cdot 222 c m^{2} )
( mathrm{D} cdot 242 mathrm{cm}^{2} )
12
25 The area of the region bounded by ( x= )
( mathbf{0}, boldsymbol{y}=mathbf{0}, boldsymbol{x}=mathbf{2}, boldsymbol{y}=mathbf{2}, boldsymbol{y} leq boldsymbol{e}^{boldsymbol{x}} ) and
( boldsymbol{y} geq ln boldsymbol{x}, ) is
( mathbf{A} cdot 6-4 ln 2 )
B. 4 ellп ( 2-2 )
( c cdot 2 ln 2 )
D. ( 6-2 ) ellп 2
12
26 Prove that the height of the cylinder of
maximum volume that can be inscribed
in a sphere of radius ( R ) is ( frac{2 R}{sqrt{3}} . ) Also find the maximum volume
12
27 The line ( 3 x+2 y=13 ) divides the area
enclosed by the curve ( 9 x^{2}+4 y^{2}- )
( 18 x-16 y-11=0 ) in two parts Find
the ratio of the larger area to the smaller area
A ( cdot frac{3 pi+2}{pi-2} )
в. ( frac{3 pi-2}{pi+2} )
c. ( frac{pi+2}{pi-2} )
D. ( frac{pi-2}{pi+2} )
12
28 Sketch the region
( left{(x, y): 9 x^{2}+4 y^{2}=36right} ) and find the
area of the region enclosed by it, using integration.
12
29 Find the area of the region common to
be circle ( x^{2}+y^{2}=9 ) and the parabola
( boldsymbol{y}^{2}=mathbf{8} boldsymbol{x} )
12
30 fthe area of the region bounded by point ( (x, y) ) satisfying the condition
( left{(x, y) ; 0 leq y leq x^{2}+1,0 leq y leq x+1right. )
12
31 Find area enclosed by ( |x|+|y| leq 3 ) and
( x y geq 2 )
( mathbf{A} .5-2 log 4 )
B. ( 3-2 log 4 )
c. ( 3-log 4 )
D. ( 3-2 log 7 )
12
32 For the curve ( f(x)=frac{1}{1+x^{2}}, ) let two
points on it be ( boldsymbol{A}(boldsymbol{alpha}, boldsymbol{f}(boldsymbol{alpha})), boldsymbol{B}left(-frac{1}{boldsymbol{alpha}}, boldsymbol{f}left(-frac{1}{boldsymbol{alpha}}right)right)(boldsymbol{alpha}>mathbf{0}) )
Find the minimum area bounded by
the line segments ( 0 A, O B ) and ( f(x) ) where ‘O’ is the origin.
A ( cdot frac{(pi-1)}{2} )
в. ( frac{pi}{2} )
c. ( frac{(pi-2)}{2} )
D. Maximum area is always infinitt
12
33 ( * )
( * )
( * )
( k )
12
34 Which of the following is true regarding the symmetry of the function: ( f(x)= ) ( boldsymbol{x}^{boldsymbol{5}}+boldsymbol{x}^{boldsymbol{3}}+boldsymbol{3} )
A. ( f(x)=c )
B. Symmetric about x-axis
c. Its an odd function
D. None of these
12
35 The area common to the cardioids ( r= )
( boldsymbol{a}(mathbf{1}+cos boldsymbol{theta}) ) and ( boldsymbol{r}=boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) ) is:
( ^{A} cdotleft(frac{3 pi}{2}+4right) a^{2} )
в. ( left(frac{3 pi}{2}-4right) a^{2} )
c. ( left(frac{pi}{2}+4right) a^{2} )
D ( cdotleft(frac{pi}{2}-4right) a^{2} )
12
36 Area bounded by the curves satisfying the conditions ( frac{x^{2}}{25}+frac{y^{2}}{36} leq 1 leq frac{x}{5}+frac{y}{6} )
is given by
A ( cdot 15left(frac{pi}{2}+1right) ) sq.units
B. ( frac{15}{4}left(frac{pi}{2}-1right) ) sq.units
c. ( 30(pi-1) ) sq.unit
D. ( frac{15}{2}(pi-2) ) sq.unit
12
37 If the area enclosed by the parabolas ( y=a-x^{2} ) and ( y=x^{2} ) is ( 18 sqrt{2} ) sq.
units Find the value of ‘a’
A ( . a=-9 )
B. ( a=6 )
( mathbf{c} cdot a=9 )
D. ( a=-6 )
12
38 Find the area of the region enclosed by
the parabola ( x^{2}=y, ) the line ( y=x+2 )
and ( x ) -axis.
12
39 The area of the curve ( y^{2}=(7-x)(5+ )
( x ) ) above ( x ) -axis and between the
ordinates ( x=-5 ) and ( x=1 ) is ( n pi ). Find
( boldsymbol{n} )
12
40 The area bounded by the parabola ( y^{2}= )
( 4 x ) and the line ( y=2 x-4 )
A. 9 sq. units
B. 5 sq. units
c. 4 sq. units
D. 2 sq. units
12
41 Find the area bounded by ( y=x+sin x )
and its inverse between ( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}= )
( 2 pi )
A .2
B. 4
( c .6 )
D. 8
12
42 Find the area bounded by the curve ( y= ) ( cos , x- ) axis and the ordinates ( x=0 )
and ( x=2 pi )
12
43 Using integration, find the area bounded by the curve ( x^{2}=4 y ) and the
line ( x=4 y-2 )
12
44 Find area bounded by ( x^{2}+y^{2} leq )
( 2 a x, y^{2} geq a x, x geq 0, y leq 0 )
12
45 ( # )
( * )
( k )
( k )
12
46 Area bounded by the curves ( y=x^{2} ) and
( y=2-x^{2} ) is
A ( cdot frac{8}{3} ) sq. units
B. ( frac{3}{8} ) sq. units
c. ( _{-5 text {funits }}^{text {s }} )
D. None of these
12
47 Example 2.7 Using the method of integration, show that
the volume of a right circular cone of base radius r and height
his V = -rah.
3
12
48 The parabola ( y^{2}=4 x+1 ) divides the
( operatorname{disc} x^{2}+y^{2} leq 1 ) into two regions with
( operatorname{areas} A_{1} ) and ( A_{2} . ) Then ( left|A_{1}-A_{2}right| ) equals
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
c.
D. ( frac{pi}{3} )
12
49 The area bounded by ( x^{2}=4 a y ) and ( y= )
( 2 a ) is?
A ( cdot frac{16 sqrt{2} a^{2}}{3} )
в. ( frac{16 a^{2}}{3} )
c. ( frac{8 a^{2}}{3} )
D. ( frac{8 sqrt{2} a^{2}}{3} )
12
50 Find the area bounded on the right by
the line ( x+y=2, ) on the left by the
parabola ( y=x^{2} ) and below by the ( x- )
axis.
12
51 The area of the region bounded by the
curve ( boldsymbol{x}=boldsymbol{y}^{2}-boldsymbol{2} ) and ( boldsymbol{x}=boldsymbol{y} ) is
A ( cdot frac{9}{4} )
B. 9
c. ( frac{9}{2} )
D.
12
52 The area of the smaller part of the circle ( x^{2}+y^{2}=a^{2}, ) cut off by the line ( x=frac{a}{sqrt{2}} )
is given by:
( ^{text {A }} cdot frac{a^{2}}{2}left(frac{pi}{2}+1right) )
в. ( frac{a^{2}}{2}left(frac{pi}{2}-1right) )
c. ( a^{2}left(frac{pi}{2}-1right) )
D. None of these
12
53 Find the area between the two
parabolas ( boldsymbol{y}^{2}=boldsymbol{x}, boldsymbol{X}^{2}=boldsymbol{y} )
12
54 41. Find the area of the region bounded by the curves y = x², y
= 12 – xº|and y=2, which lies to the right of the line x=1.
(2002 – 5 Marks)
12
55 Let ( boldsymbol{y}=boldsymbol{g}(boldsymbol{x}) ) be the inverse of ( mathbf{a} )
bijective mapping ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} quad boldsymbol{f}(boldsymbol{x})= )
( 3 x^{3}+2 x . ) The area bounded by graph of
( g(x), ) the axis and the ordinate at ( x=5 )
is
A ( cdot frac{5}{4} )
B. ( frac{7}{4} )
( c cdot frac{9}{4} )
D. ( frac{13}{4} )
12
56 If
( f(x) ) is monotonic in ( (a, b) ) then prove that the area bounded by the ordinates
at ( boldsymbol{x}=boldsymbol{a}: boldsymbol{x}=boldsymbol{b}: boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{y}= )
( f(c), c epsilon(a, b) ) is minimum when ( c= ) ( frac{a+b}{2} )
Hence if the area bounded by the graph of ( f(x)=frac{x^{3}}{3}-x^{2}+a, ) the straight
lines ( x=0, x=2 ) and the ( x ) -axis is
minimum then find the value or ‘a’.
( A cdot frac{2}{3} )
B. ( frac{2}{5} )
( c cdot frac{7}{3} )
D. ( frac{4}{3} )
12
57 Find the area of the region bounded by the curves ( x=frac{1}{2}, x=2, y=log x ) and
( boldsymbol{y}=mathbf{2}^{boldsymbol{x}} )
( ^{mathbf{A}} cdot frac{4-sqrt{2}}{log 2}-frac{5}{2} log 2+frac{3}{2} s q cdot u n i t s )
B. ( frac{4+sqrt{2}}{log 2}-frac{3}{2} log 2+frac{5}{2} ) sq.units
( ^{mathrm{c}} cdot frac{4-sqrt{2}}{log 2}-frac{3}{2} log 2+frac{5}{2} s q cdot u n i t s )
D. ( frac{4+sqrt{2}}{log 2}-frac{5}{2} log 2+frac{3}{2} ) sq.units
12
58 Let S be the area of the region enclosed by y = e*
v=0, x=0 and x= 1; then
(2012)
(a) Sz-
(6) 821-1
A
V
+-
hot otiofith
12
59 The area bounded by the line ( x=1 ) and the curve ( sqrt{frac{boldsymbol{y}}{boldsymbol{x}}}+sqrt{frac{boldsymbol{x}}{boldsymbol{y}}}=mathbf{4} ) is
A ( .2 sqrt{3} )
B. ( sqrt{3} )
c. ( 3 sqrt{2} )
D. ( 4 sqrt{3} )
12
60 Area bounded by the curves ( frac{y}{x}=log x ) and ( frac{boldsymbol{y}}{mathbf{2}}=-boldsymbol{x}^{2}+boldsymbol{x} ) equals:
A. 7/12
B. 12/7
( c cdot 7 / 6 )
D. 6/7
12
61 Find the area under the curve ( boldsymbol{y}= )
( left(x^{2}+2right)^{2}+2 x ) between the lines ( x=0 )
( x=2 ) and the ( X ) -axis
12
62 34. Let the straight line x = b divide the area enclosed by
y=(1-x)2 , y=0, andx=0 into two parts R, (0 < x < b) and
Rz (b < x < 1) such that R1 – R2 = 2. Then b equals (2011)
12
63 30.
The area of the plane region bounded by the curves
x+2y2 = 0 and x+3y2 = lis equal to
[2008]
12
64 Determine the area of the
region enclosed by the curve ( boldsymbol{y}= ) ( sqrt{x+1} ) in [0,4] and the ( x ) -axis and the
lines ( boldsymbol{x}=mathbf{0}, boldsymbol{x}=mathbf{4} )
12
65 The area bounded by ( y=cos x, y= )
( boldsymbol{x}+mathbf{1}, boldsymbol{y}=mathbf{0} ) is
A ( cdot frac{3}{2} )
B. ( frac{2}{3} )
( c cdot frac{1}{2} )
D. ( frac{5}{2} )
12
66 For ( x, t in R ) let
( boldsymbol{p}_{1}(boldsymbol{x})=(sin t) boldsymbol{x}^{2}-(2 cos boldsymbol{t}) boldsymbol{x}+sin boldsymbol{t} )
be a family of quadratic polynomials in ( x ) with variable co efficients. Let ( A(t)= ) ( int_{0}^{1} p_{t}(x) d x . ) Which of the following
statements are true?
(I) ( A(t)<0 ) for all ( t )
(II) ( A(t) ) has infinitely many critica points.
(III) ( A(t)=0 ) for infinitely many t.
(IV) ( A^{prime}(t)<0 ) for all ( t )
A. (I) and (II) only
B. (II) and (III) only
C. (III) and (IV) only
( D cdot(I V) ) and (1) only
12
67 The area bounded by the curve ( y= ) ( sin ^{-1} x ) and ( operatorname{lines} x=0,|y|=frac{pi}{2} ) is 2
sq.unit
A . True
B. False
12
68 Area bounded by the curves ( boldsymbol{y}= )
( e^{x}, y=e^{-x} ) and the straight line ( x=1 )
is (in sq units)
A ( cdot_{e+frac{1}{e}} )
B. ( e+frac{1}{e}+2 )
c. ( _{e+frac{1}{e}-2} )
D. ( _{e-frac{1}{e}+2} )
E ( e-frac{1}{e} )
12
69 The area bounded by ( y=cos x, y= )
( x+1 ) and ( y=0 ) in the second quadrant
is
A ( cdot frac{3}{2} ) sq. units
B. 2 sq. units
c. 1 sq. unit
D. ( frac{1}{2} ) sq, units
12
70 The area bounded by the curves ( x^{2}+ )
( y^{2} leq 8 ) and ( y^{2} geq 4 x ) lying in the first
quadrant is not equal to
A ( cdot 32left(frac{pi}{8}-frac{1}{3}right) )
в. ( frac{32}{3}left(frac{3 pi}{8}-1right) )
c. ( _{4 pi}-frac{32}{3} )
D. ( frac{1}{3}(12 pi-32) )
12
71 The area bounded by the curve ( x^{2}=4 y )
and straight line ( x=4 y-2 ) is
( A cdot frac{3}{8} )
B.
( c cdot frac{7}{8} )
D.
12
72 The larger area bounded between the
curve ( |boldsymbol{x}|^{2}+|boldsymbol{y}|^{2}=mathbf{1} ) and the line ( boldsymbol{x}+ )
( y=1 ) is
( ^{mathrm{A}} cdotleft(frac{pi}{4}-frac{1}{2}right) ) sq. unit
B. ( left(frac{pi}{4}+1right) ) sq. unit
c. ( left(frac{3 pi}{4}+frac{1}{2}right) ) sq. unit
D ( cdotleft(frac{3 pi}{4}-frac{1}{2}right) ) sq. unit
12
73 Find the area of the region bounded by the ellipse ( frac{x^{2}}{4}+frac{y^{2}}{9}=1 ) 12
74 If the area of the region bounded by the curves, ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=frac{mathbf{1}}{boldsymbol{x}} ) and the lines
( boldsymbol{y}=mathbf{0} ) and ( boldsymbol{x}=boldsymbol{t}(boldsymbol{t}>mathbf{1}) ) is ( mathbf{1} ) sq. unit
then ( t ) is equal to?
A ( cdot frac{4}{3} )
В. ( e^{2 / 3} )
( c cdot frac{3}{2} )
( mathbf{D} cdot e^{3 / 2} )
12
75 Find the area of the region bounded by
the curve ( x y=c^{2}, ) the ( X ) -axis, and the
( operatorname{lines} x=c, x=2 c )
12
76 Find the area bounded by the curve ( y= )
( 2 x-x^{2} ) and the straight line ( y=-x )
12
77 1.
The area (in sq. units) bounded by the parabola y=x2-1.
the tangent at the point (2, 3) to it and the y-axis is:
[JEE M 2019-9 Jan (M)
(a)
wloo
(b)
(c) 56
3
12
78 The area of region bounded by curve ( y=cos 2 x, ) line ( x=0 ) and ( x=frac{pi}{3} ) is
A ( cdot frac{2-sqrt{3}}{4} )
B. ( frac{sqrt{3}}{4} )
c. ( frac{4-sqrt{3}}{4} )
D. ( frac{sqrt{3}-4}{4} )
12
79 The parabola ( y^{2}=4 x ) and ( x^{2}=4 y )
divide the square region bounded by the
( operatorname{lines} x=4, y=4 ) and the coordinate
axes. If ( S_{1}, S_{2}, S_{3} ) are the areas of these
parts numbered from top to bottom
respectively, then This question has multiple correct options
A ( . S_{1}: S_{2} equiv 1: 1 )
B. ( S_{2}: S_{3} equiv 1: 2 )
c. ( S_{1}: S_{3} equiv 1: 1 )
( mathbf{D} cdot S_{1}:left(S_{1}+S_{2}right) equiv 1: 2 )
12
80 The area bounded by ( boldsymbol{x}=boldsymbol{a} cos ^{3} boldsymbol{theta}, boldsymbol{y}= )
( a sin ^{3} theta ) is:
A ( cdot frac{3 pi a^{2}}{16} )
в. ( frac{3 pi^{2}}{8} )
c. ( frac{3 pi a^{2}}{32} )
D. ( 3 pi a^{2} )
12
81 The points at which the tangents to the curve ( y=x^{3}-12 x+18 ) are parallel to
x-axis are?
A ( cdot(2,-2),(-2,-34) )
в. (2,34),(-2,0)
c. (0,34),(-2,0)
D. (2,2),(-2,34)
12
82 What is the smallest distance between
the point (-2,-2) and a point on the circumference of the circle given by
( (x-1)^{2}+(y-2)^{2}=4 ? )
( A cdot 3 )
B. 4
( c .5 )
D. 6
E. 7
12
83 Area bounded by the parabola ( y^{2}=x- )
3 and ( x=5, ) is ( dots . s q ) units
( A cdot frac{8 sqrt{2}}{3} )
B. ( 4 sqrt{2} )
( mathrm{c} cdot 3 sqrt{2} )
D. ( 5 sqrt{2} )
12
84 The ratio in which the area bounded by
the curves ( y^{2}=4 x ) and ( x^{2}=4 y )
is divided by the line ( x=1 ) is
A. 64: 49
B. 15: 34
c. 15: 49
D. None o fthese
12
85 Prove that the area enclosed between
two parabolas ( y^{2}=4 a x ) and ( x^{2}=4 a y )
is ( frac{16 a^{2}}{3} )
12
86 31.
Consider a square with vertices at (1, 1), (-1, 1), (-1,-1)
and (1, -1). Let S be the region consisting of all points
inside the square which are nearer to the origin than to any
edge. Sketch the region S and find its area.
12
87 Using the method of integraton find the area of the region bounded by lines:
( 2 x+y=4,3 x-2 y=6 ) and ( x-3 y+ )
( 5=0 )
12
88 A square is inscribed in a circle of
radius ( 7 mathrm{cm} ). Find the maximum area of
the square.
( mathbf{A} cdot 98 )
B. 89
c. 86
D. 96
12
89 Find the area of the region bounded by the ellipse ( frac{x^{2}}{16}+frac{y^{2}}{9}=1 )
A . ( 6 pi )
в. ( 12 pi )
c. ( 18 pi )
D. ( 24 pi )
12
90 The area of the plane region bounded by
the curve ( x+2 y^{2}=0 ) and ( x+3 y^{2}=1 )
is equal to:
A ( cdot-frac{4}{3} )
B. ( frac{4}{3} )
( c cdot frac{2}{3} )
D. None of these
12
91 Area bounded by ( x^{2}=4 a y ) and ( y= )
( frac{8 a^{3}}{x^{2}+4 a^{2}} ) is :
12
92 Show that the maximum volume of the
cylinder which can be inscribed in a sphere of radius ( 5 sqrt{3} mathrm{cm} ) is ( 500 pi c m^{3} ? )
12
93 If the area enclosed between the curves
( boldsymbol{y}=boldsymbol{k} boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{k} boldsymbol{y}^{2},(boldsymbol{k}>mathbf{0}), ) is ( mathbf{1} )
square unit. Then ( k ) is?
A ( cdot frac{1}{sqrt{3}} )
в. ( frac{2}{sqrt{3}} )
c. ( frac{sqrt{3}}{2} )
D. ( sqrt{3} )
12
94 Find the equation of the curve passing
through the point ( (0,1), ) if the slope of
tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and
ordinate of the point.
12
95 Area of the region bounded by the
curves, ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}, boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} ) and the straight
line ( x=1 ) is given by
A ( cdotleft(e-e^{-1}+2right) ) sq.unit
B ( cdotleft(e-e^{-1}-2right) ) sq.unit
c. ( left(e+e^{-1}-2right) ) sq.unit
D. None of the above
12
96 The value of the integral ( int_{1}^{2} sqrt{(2 x+3)left(3 x^{2}+4right)} d x ) cannot
exceed
( A cdot sqrt{48} )
B. ( sqrt{66} )
c. ( sqrt{73} )
D. none of these
12
97 The area (in square units) bounded by
( boldsymbol{y}=boldsymbol{x} e^{|x|} ) and lines ( |boldsymbol{x}|=mathbf{1}, boldsymbol{y}=mathbf{0}, ) is
A .4
B. 6
( c .1 )
D. 2
12
98 The smaller area enclosed by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) )
where ( f(x) ) is polynomial of least degree satisfying ( left[lim _{x rightarrow 0} 1+frac{f(x)}{x^{3}}right]^{frac{1}{x}}=e )
and the circle ( x^{2}+y^{2}=2 ) above the ( x- )
axis is
A ( cdot frac{pi}{2}+frac{3}{5} )
в. ( frac{pi}{2}-frac{3}{5} )
c. ( frac{pi}{2}-frac{6}{5} )
D. None of these
12
99 The area enclosed between the
parabolas ( boldsymbol{y}^{2}=mathbf{1 6} boldsymbol{x} ) and ( boldsymbol{x}^{2}=mathbf{1 6} boldsymbol{y} ) is
A ( cdot frac{64}{3} ) sq.units
в. ( frac{256}{3} ) sq.units
c. ( frac{16}{3} ) sq.units
D. None of these
12
100 Show that the height of a closed right circular cylinder of given surface and
maximum volume, is equal to the diameter of its base.
12
101 Give possible expressions for the length and breadth of the rectangle whose area is ( 3 x^{2}-8 x+5 ) 12
102 A conic ( C ) passing through ( P(1,2) ) is
such that the slope of its tangent at any point on the conic is inversely proportional to the ordinate of that point and conic ( C ) passes through origin. If a circle touches the conic ( C ) at the
point ( boldsymbol{P}(1,2) ) and passes through the focus of the conic then its radius is-
A.
B. ( sqrt{2} )
( c cdot 2 )
D. ( sqrt{3} )
E. ( sqrt{5} )
12
103 The area of the region between the curve
( boldsymbol{y}=boldsymbol{x}^{3} ) and the lines ( boldsymbol{y}=-boldsymbol{x} ) and ( boldsymbol{y}=mathbf{1} )
is:
A. 5 sq. units
B. ( frac{4}{5} ) sq. units
c. ( frac{5}{4} ) sq. units
D. ( frac{3}{5} ) sq. units
12
104 Draw the graph of ( 2 x+y=6 ) and ( 2 x- )
( boldsymbol{y}+mathbf{2}=mathbf{0} . ) Shade the region bounded by
these lines and ( x-y . ) Find the area of
the shaded region.
12
105 12.
(1983 – 2
Sketch the region bounded by the curves y=V5-X and
y=x-1) and find its area.
(1985 – 5 Marks)
12
106 Find the area of the region bounded by
the curve ( y^{2}=4 x quad ) and the line ( x=3 )
12
107 If the area enclosed by ( y^{2}=4 a x ) and line ( y=a x ) is ( frac{1}{3} . ) sq.unit, then the area enclosed by ( y=4 x ) with same parabola is ( frac{4}{3} ) squnit
A. True
B. False
12
108 Show that the line ( frac{x}{a}+frac{y}{b}=1, ) tocuhes the curve ( y=b . e^{frac{-x}{a}} ) at the point where
the curve intersects the axis of ( y )
12
109 Let ( f(x)=x^{frac{2}{3}}, x geq 0 . ) Then the area of
the region enclosed by the curve ( y= ) ( f(x) ) and the three lines ( y=x, x=1 )
and ( x=8 )
A ( cdot frac{63}{2} )
в. ( frac{93}{5} )
c. ( frac{105}{7} )
D. ( frac{129}{10} )
12
110 The area of the figure bounded by the ( operatorname{lines} x=0, x=frac{pi}{2}, f(x)=sin x ) and
( g(x)=cos x ) is
A ( cdot 2(sqrt{2}-1) )
B. ( sqrt{3}-1 )
c. ( 2(sqrt{3}-1) )
D ( cdot 2(sqrt{2}+1) )
12
111 Find the equations corresponding to following graph:
A. ( y=-4 x-6 )
B. ( y=4 x-6 )
( c cdot y=4 x )
D. none of these
12
112 The area bounded by the curve ( y=sqrt{x} )
the line ( 2 y+3=x ) and the ( x ) -axis in
the first quadrant is
( mathbf{A} cdot mathbf{9} )
в. ( frac{27}{4} )
( c .36 )
D. 18
12
113 Find the common area enclosed by the parabolas ( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}^{2}=boldsymbol{y} ) 12
114 Area of the region bounded by ( y^{2} leq ) ( mathbf{4} boldsymbol{x}, boldsymbol{x}+boldsymbol{y} leq mathbf{1}, boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} ) is ( boldsymbol{a} sqrt{mathbf{2}}+boldsymbol{b} )
then value of ( a-b ) is?
A .4
B. 6
( c cdot 8 )
D. 12
12
115 (
UJU
16. The area of the region bounded by the curves
y = x-2), x = 1, x = 3 and the x-axis is
(a) 4 (6) 2 (c) 3 (d)
[2004]
1
12
116 Find the area of the region bounded by the curve ( y^{2}=x ) and the lines ( x= )
( 1, x=4 ) and the ( x ) -axis.
12
117 Area enclosed by the curves ( y= )
( ln x ; y=ln |x| ; y=|ln x| ) and ( y= )
( |ln | x|| ) is equal to
( A cdot 2 )
B. 4
( c cdot 8 )
D. cannot be determined
12
118 A tangent to the curve ( y=x^{2}+3 x )
passes through a point (0,-9) if it drawn at the point-
A. (-3,0)
в. (1,4)
c. (0,0)
(年. ( 0,0,0,0) )
D. (-4,4)
12
119 Using integration find the area of the following region ( (x, y):|x+2| leq y leq )
( sqrt{20-x^{2}} )
12
120 Area of the region bounded by the
( operatorname{curves} boldsymbol{y}|boldsymbol{y}| pm boldsymbol{x}|boldsymbol{x}|=mathbf{1} ) and ( boldsymbol{y}=|boldsymbol{x}| ) is:
A ( cdot frac{pi}{8} ) sq.unit
B . ( frac{pi}{4} ) sq.unit
c. ( frac{pi}{2} ) sq.unit
D. ( pi ) sq.unit
12
121 Find the area of the figure contained
between the parabola ( x^{2}=4 y ) and the
curve ( boldsymbol{y}=frac{boldsymbol{8}}{boldsymbol{x}^{2}+boldsymbol{4}} )
12
122 Consider the curves ( y=sin x ) and ( y= )
( cos x )
What is the area of the region bounded by the above two curves and the lines ( x=frac{pi}{4} ) and ( x=frac{pi}{2} ? )
B. ( sqrt{2}+1 )
( c cdot 2 sqrt{2} )
( D )
12
123 Area enclosed between the curves ( y= )
( 8-x^{2} ) and ( y=x^{2}, ) is:
A . ( 32 / 3 )
B. ( 64 / 3 )
c. ( 30 / 4 )
D.
12
124 f ( boldsymbol{y}=boldsymbol{x}^{2}+boldsymbol{x}, ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
125 36. Let C, and C, be the graphs of the functions y = x2 and
y = 2x, 0 < x < 1 respectively. Let C, be the graph of a
function y=f(x), 0 SX S1,f(0) = 0. For a point P on C, let
the lines through P, parallel to the axes, meet C, and C, at Q
and R respectively (see figure.) If for every position of P (on
(), the areas of the shaded regions OPQ and ORP are
12
126 The area between the curves ( y=sqrt{x} )
and ( boldsymbol{y}=boldsymbol{x}^{3} ) is
A ( cdot frac{1}{12} ) sq. units
в. ( frac{5}{12} ) sq. units
c. ( frac{3}{5} ) sq. units
D. ( frac{4}{5} ) sq. units
12
127 The subnormal at any point on the curve
( x y^{n}=a^{n+1} ) is constant for
( mathbf{A} cdot n=0 )
B . ( n=1 )
c. ( n=-2 )
D. ( n=2 )
12
128 Find area between:
( boldsymbol{x}=mathbf{2} boldsymbol{y}-boldsymbol{y}^{2} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{y}=boldsymbol{2}+boldsymbol{x} )
12
129 toppr
Q Type your question_
or ( x ) between the oralnates ( x=a ) メ ( x= )
( b, ) is given by definite integral
( int_{a}^{b} y d x ) or ( int_{a}^{b} f(x) d x ) and the area
bounded by the curve ( x=f(y), ) the axis
of ( y & ) two abscissae ( y=c & y=d ) is
given by ( int_{c}^{d} x d y ) or ( int_{c}^{d} f(x) d y . ) Again if
we consider two curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}= )
( g(x) ) where ( f(x) geq g(x) ) in the interval
( [a, b] ) where ( x=a & x=b ) are the
points of intersection of these two
curves Shown by the graph given
Then area bounded by these two curves
is given by
( int_{a}^{b}[boldsymbol{f}(boldsymbol{x})-boldsymbol{g}(boldsymbol{x})] boldsymbol{d} boldsymbol{x} )
On the basis of above information
answer the following questions.
The area bounded by parabolas ( boldsymbol{y}= )
( x^{2}+2 x+1 & y=x^{2}-2 x+1 ) and the
line ( y=frac{1}{4} ) is equal to
( mathbf{A} )
square unit
B. ( overline{3} ) square unit
( c )
( overline{2} ) square unit
D. 2 square unit
12
130 Find the smaller area enclosed by the
circle ( x^{2}+y^{2} ) and the line ( x+y=2 )
12
131 Draw a rough sketch of the curve ( y^{2}= )
( 4 x ) and find the area of the region
enclosed by the curve and the line ( y=x )
12
132 The area between the curves ( y=x^{2} ) and
( boldsymbol{y}=frac{2}{1+x^{2}} ) is
A ( cdot pi-frac{1}{3} )
B . ( pi-2 )
c. ( pi-frac{2}{3} )
D. ( pi+frac{2}{3} )
12
133 Set up an equation of a tangent to the graph of the following function. A cone is circumscribed about a sphere
of radius R. The vertex angle in the
axial section of the cone is ( 2 alpha ). Find the
area of the axial section of the cone.
At what value of ( alpha ) is the area of the
cone the least?
12
134 The area (in sq.units) of the region
( left{(x, y): y^{2} geq 2 x text { and } x^{2}+y^{2} leqright. )
( 4 x, x geq 0, y geq 0 ) is
A ( cdot pi-frac{8}{3} )
B. ( _{pi-frac{4 sqrt{2}}{3}} )
( ^{mathrm{C}} cdot frac{pi}{2}-frac{2 sqrt{2}}{3} )
D. ( pi-frac{4}{3} )
12
135 The area in the first quadrant bounded by the curves ( x^{2}=2 y, y^{2}=2 x ) and
( x^{2}+y^{2}=3 ) is
12
136 Which of the following equations shows
the shaded region in the above figure?
A ( cdot xleft(y-frac{2}{3} xright) geq 0 )
B. ( xleft(y-frac{3}{2} xright) geq 0 )
c. ( xleft(y+frac{3}{2} xright) geq 0 )
D ( cdot xleft(y+frac{2}{3} xright) geq 0 )
E ( cdot xleft(y+frac{3}{2} xright) leq 0 )
12
137 Show that all rectangles inscribed in a fixed circle square has maximum area. 12
138 An equilateral triangle has area ( A mathrm{cm}^{2} ) A regular hexagon of maximum area is cut off from the triangle. If the area of
the hexagon is ( 320 mathrm{cm}^{2} ), then the area ( A )
is
A ( cdot 640 mathrm{cm}^{2} )
в. ( 480 mathrm{cm}^{2} )
( c cdot 600 c m^{2} )
D. ( 400 mathrm{cm}^{2} )
12
139 If the area bounded by the curve ( |boldsymbol{y}|= )
( sin ^{-1}|x| ) and ( x=1 ) is ( a(pi+b), ) then
the value ( a-b ) is:
A ( . )
B. 2
( c cdot 3 )
D. 4
12
140 The area of the region, bounded by the curves ( boldsymbol{y}=sin ^{-1} boldsymbol{x}+boldsymbol{x}(mathbf{1}-boldsymbol{x}) ) and ( boldsymbol{y}= )
( sin ^{-1} x-x(1-x) ) in the first quadrant
is
( A cdot 1 )
в. ( frac{1}{2} )
( c cdot frac{1}{3} )
D.
12
141 Find the area of the region bounded by
the following curves, the ( X ) -axis and
the given lines.
( boldsymbol{y}=boldsymbol{x}^{4}, boldsymbol{x}=mathbf{1}, boldsymbol{x}=mathbf{5} )
12
142 The triangle formed by the tangent to
the curve ( f(x)=x^{2}+b x-b ) at the
point (1,1) and the co-ordinate axes
lies in the first quadrant.lf its area is 2 sq.unit, then the value of ( b ) is:
A. -3 sq.unit
B. – 2 sq.unit
c. -1 sq.unit
D. 0 sq.unit
12
143 Maximum area of rectangle whose two
sides are ( boldsymbol{x}=boldsymbol{x}_{0}, boldsymbol{x}=boldsymbol{pi}-boldsymbol{x}_{0} ) and which
is inscribed in a region bounded by ( boldsymbol{y}= )
( sin x ) and ( x- ) axis is obtained, when ( x_{0} in )
12
144 Show that the differential equation of the family of circles having their centre
at the origin and radius ( a ) is
[
boldsymbol{x}+boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0}
]
12
145 The area bounded by the curves ( y^{2}=4 x )
and ( x^{2}=4 y ) is :
A ( cdot frac{32}{3} )
B. ( frac{16}{3} )
( c cdot frac{8}{3} )
D.
12
146 Find the area of the region enclosed between the two circles ( x^{2}+y^{2}=1 & )
( (x-1)^{2}+y^{2}=1 )
A ( cdot frac{pi}{6}-frac{sqrt{3}}{2} ) squnits
B. ( frac{pi}{3}-frac{sqrt{3}}{2} ) sq.units
c. ( frac{pi}{6}-frac{sqrt{3}}{4} ) sq.units
D. ( frac{pi}{3}-frac{sqrt{3}}{4} ) sq.units
12
147 24.
Sketch the region bounded by the curves y = x- and
2
y=-
9. Find the area.
1+x2
(1992 – 4 Marks)
12
148 The angle of intersection between the
curves ( boldsymbol{x}^{2}=mathbf{4}(boldsymbol{y}+mathbf{1}) ) and ( boldsymbol{x}^{2}=-mathbf{4}(boldsymbol{y}+ )
1) is.
A ( cdot frac{pi}{6} )
B. ( frac{pi}{4} )
c. 0
D.
12
149 Area common to the curves ( y^{2}=a x )
and ( x^{2}+y^{2}=4 a x ) is equal to
A ( cdot(9 sqrt{3}+4 pi) frac{a^{2}}{3} )
B . ( (9 sqrt{3}+4 pi) a^{2} )
c. ( (9 sqrt{3}-4 pi) frac{a^{2}}{3} )
D. None of these
12
150 The area enclosed between the curves
( x^{2}=y ) and ( y^{2}=x ) is equal to
This question has multiple correct options
A ( cdot frac{1}{3} cdot ) sq.unit
B ( cdot 2 int_{0}^{1}left(x-x^{2}right) d x )
C cdot area of the region ( left{(x, y): x^{2} leq y leq|x|right} )
D. none of the above
12
151 The area of the smaller part bounded by the semicircle ( boldsymbol{y}=sqrt{mathbf{4}-boldsymbol{x}^{2}}, boldsymbol{y}=boldsymbol{x} sqrt{mathbf{3}} )
and x-axis is
A ( cdot frac{pi}{3} )
в. ( frac{2 pi}{3} )
c. ( frac{4 pi}{3} )
D.
12
152 The area bounded by parabola ( boldsymbol{y}^{2}=boldsymbol{x} )
straight line ( y=4 ) and ( y-a x i s ) is-
A ( cdot frac{16}{3} )
B. ( 7 sqrt{2} )
c. ( frac{32}{3} )
D. ( frac{64}{3} )
12
153 The ratio in which the area bounded by
the curves ( y^{2}=4 x ) and ( x^{2}=4 y )
is divided by the line ( x=1 ) is
A. 64: 49
B. 15: 34
c. 15: 49
D. None o fthese
12
154 An edge of a variable cube is increasing at the rate of ( 3 mathrm{cm} / mathrm{s} ). How fast is the
volume of the cube increasing when the
edge is ( 10 mathrm{cm} ) long?
A. ( 900 mathrm{cm}^{3} / mathrm{s} )
B. ( 920 mathrm{cm}^{3} / mathrm{s} )
( mathbf{c} .850 mathrm{cm}^{3} / mathrm{s} )
D. ( 950 mathrm{cm}^{3} / mathrm{s} )
12
155 Sketch the graph for ( y=1+ )
( mathbf{3}(log |sin boldsymbol{x}|+log |mathbf{c s c} boldsymbol{x}|) )
12
156 The area bounded by the tangent and
normal to the curve ( y(6-x)=x^{2} ) at
(3,3) and the ( x ) -axis is
( mathbf{A} cdot mathbf{5} )
B. 6
c. 15
D. 3
12
157 The area bounded by the circle ( x^{2}+ )
( y^{2}=8, ) the parabola ( x^{2}=2 y ) and the
line ( y=x ) in ( y geq 0 ) is
A ( cdot frac{2}{3}+2 pi )
в. ( frac{2}{3}-2 pi )
c. ( frac{2}{3}+pi )
D. ( frac{2}{3}-pi )
12
158 Two vertices of a rectangle are on the
positive ( mathbf{x} ) -axis. The other two vertices lie on the lines ( y=4 x ) and ( y=-5 x+6 )
Then the maximum area of the
rectangle is?
( A cdot frac{2}{3} )
B. ( frac{2}{4} )
( c cdot frac{1}{3} )
D. ( frac{4}{3} )
12
159 Find a continuous function ( f, ) where
( left(x^{4}-4 x^{2}right) leq f(x) leqleft(2 x^{2}-x^{3}right) ) such
that the area bounded by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}= )
( x^{4}-4 x^{2}, y ) -axis, and the line ( x=t )
where ( (0 leq t leq 2) ) is ( k ) times the area
bounded by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}=mathbf{2} boldsymbol{x}^{2}-boldsymbol{x}^{3}, mathbf{y} )
axis, and line ( boldsymbol{x}=boldsymbol{t} text { (where } mathbf{0} leq boldsymbol{t} leq mathbf{2}) )
12
160 Find area of circle ( 4 x^{2}+4 y^{2}=9 ) which
is interior to the parabola ( x^{2}=4 y )
12
161 28. The area
area bounded by the parabolas y = (x + 1)2 and
= (x – 1)2 and the line y = 1/4 is
(2005)
(b) 1/6 sq. units
(d) 1/3 sq. units
(a) 4 sq. units
(c) 4/3 sq. units
12
162 Of all the closed right circular
cylindrical cans of volume ( 128 pi c m^{3} )
find the dimensions of the can which
has minimum surface area.
12
163 38. L
→R (the set of all real number) be a po
number) be a positive,
non-constant and differentiable function such that
f'(x) <2f(x) and f
1. Then the value of f(x) dx lies
– 1/2
(JEE Adv. 2013)
(b) (e-1, 2e-1)
in the interval
(a) (2e-1,2e)
12
164 he line y = mx
For which of the following values of m, is the area of
region bounded by the curve y = x – x and the line y =
equals 9/2?
(1999 – 3 Marko
(a) – 4 (6) – 2 (c) 2
(d)
4
12
165 Let ‘a’ be a positive constant number.
Consider two curves ( C_{1}: y=e^{x}, C_{2} )
( y=e^{a-x} . ) Let ( S ) be the area of the part
surrounded by ( C_{1}, C_{2} ) and the ( y ) -axis,
then
A ( cdot lim _{a rightarrow infty} S=1 )
B. ( lim _{a rightarrow 0} frac{S}{a^{2}}=frac{1}{4} )
C. Range of ( operatorname{sis}(0, infty) )
D. S(a) is neither odd nor even
12
166 What is the following parabola’s axis of
symmetry?
( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+mathbf{5} )
A. ( x=-2 )
B. ( x=2 )
c. ( y=-2 )
D. None of these
12
167 Draw the graph of the linear equation ( 4 x+y=6 . ) At what points the graph of the equation cuts the ( x ) -axis and the ( y ) -axis
? Find area bounded by this line and coordinate axes.
12
168 Find the area of the region ( {(x, y) )
( left.boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4}, boldsymbol{x}+boldsymbol{y} geq mathbf{2}right} )
( mathbf{A} cdot pi-2 )
B. ( pi-1 )
c. ( 2 pi-2 )
D. ( 4 pi-2 )
12
169 28. The area enclosed between the curves y2 = x and y = |x is
[2007]
(a) 1/6 (b) 1/3 (c) 2/3 (d) i
12
170 The area between the curves ( y=sqrt{x} )
and ( boldsymbol{y}=boldsymbol{x}^{3} ) is
A ( cdot frac{1}{12} ) sq. units
в. ( frac{5}{12} ) sq. units
c. ( frac{3}{5} ) sq. units
D. ( frac{4}{5} ) sq. units
12
171 The area of the region bounded by the
curve ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{1} ) and ( boldsymbol{y}=mathbf{2} boldsymbol{x}-mathbf{2} )
between ( x=-1 ) and ( x=2 ) is:
A. 9 sq . units
B. 12sq. units
c. 15 sq. units
D. 14sq. units
12
172 n Fig ( 31 A B C D ) is a rectangle with
diameter ( 32 m ) by ( 18 m . A D E ) is a
triangle such that ( E F perp A D ) and ( E F= )
14 ( c m . ) Calculate the area of the shaded
region.
12
173 Area bounded by the curve ( y=x^{3}, ) the
x-axis and the ordinates ( x=-2 ) and
( x=1 ) is :
12
174 The area bounded by ( y=frac{3 x^{2}}{4} ) and the
line ( 3 x-2 y+12=0 ) is:
( mathbf{A} cdot mathbf{9} )
B. 18
c. 27
D. None of these
12
175 Find the area of region lying between parabolas ( y^{2}=4 a x & x^{2}=4 a y ) where ( a> )
0
12
176 A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height ( h, ) when ( h=9 mathrm{cm} ) 12
177 Area bounded by the curves ( y=x e^{x} )
and ( y=x e^{-x} ) and the line ( |x|=1 ) is
( mathbf{A} cdot mathbf{1} )
B. ( frac{4}{e} )
( c )
D. –
12
178 Two vertices of a rectangle are on the
positive ( mathbf{x} ) -axis. The other two vertices lie on the lines ( y=4 x ) and ( y=-5 x+6 )
Then the maximum area of the
rectangle is?
( A cdot frac{2}{3} )
B. ( frac{2}{4} )
( c cdot frac{1}{3} )
D. ( frac{4}{3} )
12
179 If ( f(x)=x^{2 / 3}, x geq 0 . ) Then,the area of
the region enclosed by the curve ( y= ) ( f(x) ) and the three lines ( y=x, x=1 )
and ( x=8 ) is
( ^{mathrm{A}} cdot frac{63}{2} )
в. ( frac{93}{5} )
( ^{c} cdot frac{105}{7} )
D. ( frac{129}{10} )
12
180 Consider two curves ( C_{1}: y=frac{1}{x} ) and
( C_{2}: y=ln x ) on the ( x y ) plane. Let ( D_{1} )
denotes the region surrounded by
( C_{1}, C_{2} ) and the line ( x=1 ) and ( D_{2} )
denotes the region surrounded by
( C_{1}, C_{2} ) and the line ( x=a . ) If ( D_{1}=D_{2} )
then the sum of logarithm of possible values of ( a ) is:
12
181 The curves ( y=x^{2}-1, y=8 x-x^{2}-9 )
at
A. Intersect at right angles at (2,3)
B. Touch each other at (2,3)
c. Do not intersect at (2,3)
D. Intersect at an angle ( frac{pi}{3} )
12
182 Find the area bounded by the curve ( y= )
( x^{2}+x+1 ) and tangent to it at (1,3)
from ( boldsymbol{x}=-mathbf{1} ) to ( boldsymbol{x}=mathbf{1} )
A ( cdot frac{2}{3} )
в. ( frac{5}{3} )
c. ( -frac{2}{3} )
D. ( frac{1}{3} )
12
183 The area bounded by the curves ( y= )
( log _{e} x ) and ( y=left(log _{e} xright)^{2} ) is
( mathbf{A} cdot 3-e )
B . ( e-3 )
C ( cdot frac{1}{2}(3-e) )
D. ( frac{1}{2}(e-3) )
12
184 If the ( x ) -axis divide the area of the region
bounded by the parabolas ( y=4 x-x^{2} )
and ( y=x^{2}-x ) in the ratio of ( a: b ) then
( a b ) is equal to
12
185 Find the area included between the
parabolas ( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}=boldsymbol{3}-boldsymbol{2} boldsymbol{y}^{2} )
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D.
12
186 The area of the portion of the circle
( x^{2}+y^{2}=1, ) which lies inside the
parabola ( boldsymbol{y}^{2}=mathbf{1}-boldsymbol{x}, ) is
A ( cdot frac{pi}{2}-frac{2}{3} )
B. ( frac{pi}{2}+frac{2}{3} )
c. ( frac{pi}{2}-frac{4}{3} )
D. ( frac{pi}{2}+frac{4}{3} )
12
187 Find the area cut off from the parabola
( 4 y=3 x^{2} ) by the straight line ( 2 y= )
( 3 x+12 )
A. 25 sq.units
B. 27sq.units
c. ( 36 s q ).units
D. 16 sq.units
12
188 Find the area enclosed between the
boundry of circle ( x^{2}+y^{2}=1 ) and the
line ( boldsymbol{x}+boldsymbol{y}=1 ) lying in the first
quadrant.
12
189 If the area of the closed figure bounded by the following curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2}- )
( boldsymbol{x}, boldsymbol{y}=mathbf{0} ) is k. Find ( 6 mathbf{k} )
12
190 The area enclosed by the curves ( y= )
( sin x+cos x ) and ( y=|cos x-sin x| ) over
the interval ( left[mathbf{0}, frac{pi}{2}right] ) is
A. ( 4(sqrt{2}-1) )
B. ( 2 sqrt{2}(sqrt{2}-1) )
c. ( 2(sqrt{2}+1) )
D. ( 2 sqrt{2}(sqrt{2}+1) )
12
191 14.
0 — –ww van
Find the area bounded by the curves, x2 + y2 = 25,
4y=14 – x2 | and x = 0 above the x-axis. (1987 – 6 Marks)
12
192 48. Let g(x) = cos x?,f(x)= x, and a, Ba<B) be the
roots of the quadratic equation 18×2 – 97x + 12 = 0 . Then
the area (in sq. units) bounded by the curve y =(gof)(x)
and the lines x = Q,x = B and y = 0, is: [JEE M 2018||
(a) }(V3+1)
(0) 1 1 (2-1)
(b) (V3 – V2)
(d) 1 2 (3-1)
12
193 The area of the region bounded by the parabola ( (boldsymbol{y}-mathbf{2})^{2}=(boldsymbol{x}-mathbf{1}), ) the
tangent to the parabola at the point (2,3) and the X-axis is?
( A cdot 3 )
B. 6
( c .9 )
D. 12
12
194 The area of the region bounded by ( y^{2}= ) ( x ) and ( x=36 ) is divided in the ratio 1: 7 by
the line ( x=a ). then a equals-
A. 7
B. 8
( c cdot 9 )
D.
12
195 A wire of length ( 28 mathrm{cm} ) is to be cut into 2 pieces, 1 piece is to be made into a
square ( & ) the other one into a circle
what should be the length of 2 piece so that the combined area of the square ( & ) the circle is min?
12
196 Find the area of the shaded region 12
197 The area bounded by the curves ( x= )
( a cos ^{3} t, y=a sin ^{3} t ) is
A ( cdot frac{3 pi a^{2}}{8} )
в. ( frac{3 pi a^{2}}{16} )
c. ( frac{3 pi a^{2}}{32} )
D. None of the above
12
198 Let ( S(alpha)=left{(x, y): y^{2} leq x, 0 leq x leq alpharight} )
and ( A(alpha) ) is area of the region ( S(alpha) . ) If
for a ( lambda, 0<lambda<4, A(lambda): A(4)=2: 5 )
then ( lambda ) equals
( ^{mathrm{A}} cdotleft(frac{4}{25}right)^{frac{1}{3}} )
( ^{mathrm{B}} 4left(frac{4}{25}right)^{frac{1}{3}} )
( ^{c} cdotleft(frac{2}{5}right)^{frac{1}{3}} )
( ^{mathrm{D}} 4left(frac{2}{5}right)^{frac{1}{3}} )
12
199 The area enclosed between the curves
( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}^{2}(boldsymbol{a}>boldsymbol{0}) ) is
1 sq.unit. then ( a= )
A ( cdot frac{1}{sqrt{3}} )
в. ( frac{2}{sqrt{3}} )
c. ( frac{4}{sqrt{3}} )
D. ( sqrt{3} )
12
200 The area bounded by the curve ( y= )
( f(x), ) above the ( x ) -axis, between ( x=a )
and ( boldsymbol{x}=boldsymbol{b} ) is:
( ^{mathrm{A}} cdot int_{f(a)}^{b} y d y )
B. ( int_{a}^{f(b)} x d x )
( ^{c} cdot int_{a}^{b} x d y )
D. ( int_{a}^{b} y d x )
12
201 Find the area bounded by ( boldsymbol{y}= ) ( cos ^{-1} x, y=sin ^{-1} x ) and ( y- ) axis
A ( cdot(2-sqrt{2}) ) sq. units
B ( cdot(sqrt{2}-2) ) sq. units
c. ( 2 sqrt{2} ) sq. units
D. ( sqrt{2} ) sq. units
12
202 : The area bounded by ( x=2 cos theta, y= )
( 3 sin theta ) is ( 36 pi ) sq. units.
II: The area bounded by ( x=2 cos theta, y= )
( 2 sin theta ) is ( 4 pi ) sq.units.
Which of the above statement is
correct?
A. Onlyı
B. Only II
c. Both I and II
D. Neither I nor II.
12
203 The value of ( c ) for which the area of the
figure bounded by the curve ( y=8 x^{2}- )
( x^{5}, ) the straight lines ( x=1 ) and ( x=c ) and the ( x- ) axis is equal to ( frac{16}{3} ) is
( A cdot 2 )
B . ( sqrt{8-sqrt{17}} )
( c .3 )
D. –
12
204 The area of the region bounded by the
curves ( boldsymbol{y}=|boldsymbol{x}-mathbf{1}| ) and ( boldsymbol{y}=mathbf{3}-|boldsymbol{x}| ) in
square units is
12
205 53.
The area (in sq. units) of the region
A={(x, y):xSy<x+2) is: (JEEM 2019-9 April (M)]
10
9
31
(a) ž (b) (c) (d) –
13
12
206 Find the area enclosed by the curves ( boldsymbol{y}=bmod (boldsymbol{x}-mathbf{1}) ) and ( boldsymbol{y}=boldsymbol{operatorname { m o d }}(boldsymbol{x}- )
1) +1
12
207 I: The area bounded by the line ( y=x )
and the curve ( y=x^{3} ) is ( 1 / 2 ) sq. units.
II: The area bounded by the curves ( boldsymbol{y}= )
( x^{3} ) and ( y=x^{2} ) and the ordinates ( x=1 )
( x=2 ) is ( frac{7}{12} ) sq. units.
Which of the above statement is
correct?
A. onlyı
c. Both I and II
D. Neither I nor II.
12
208 The area bounded by ( boldsymbol{y}=mathbf{2}- ) ( |2-x|, y=frac{3}{|x|} ) is
( ^{text {A } cdot frac{5-4 ln 2}{3} . text { sq.unit }} )
B. ( frac{2-ln 3}{2} . ) sq.unit
c. ( frac{4-3 ln 3}{2} ). sq.unit
D. none of these
12
209 The area bounded by ( frac{|boldsymbol{x}|}{boldsymbol{a}}+frac{|boldsymbol{y}|}{boldsymbol{b}}=mathbf{1} )
where ( a>0 ) and ( b>0 ) is
A ( cdot frac{1}{2} a b )
в. ( a b )
c. ( 2 a b )
D. ( 4 a b )
12
210 23. The area bounded by the curves y= Vx, 2y+3 = x and
x-axis in the 1st quadrant is
(2003)
(a) 9 (6) 27/4 (c) 36 (d) 18
12
211 3.
Area of the region bounded by the curve y = er and lines
x= 0 and y= e is
(2009)
(a) e-1
(b) ſ In (e+1- y) dy
(C) e-jedx
(2) Singay
12
212 33.
The area bounded by the curves y = cos x and y = sin x
between the ordinates x = 0 and x = –
[2010]
(a) 412+2 (6) 42-1 (c) 4V2+1 (d) 4/2–2
12
213 The area bounded by the curves ( y= )
( -sqrt{-x} ) and ( x=-sqrt{-y} ) were ( x, y leq 0 )
This question has multiple correct options
A. Can not be determined
B. is ( 1 / 3 )
( c cdot ) is ( 2 / 3 )
D. is same as that of the figure by the curves ( y= ) ( sqrt{-x} ; x leq 0 ) and ( x=sqrt{-y} ; y leq 0 )
12
214 Let ( f ) and ( g ) be continuous function on
( a leq x leq b ) and ( operatorname{set} p(x)=max )
( {f(x), g(x)} ) and ( q(x)=min )
( {f(x), g(x)}, ) the area bounded by the
curves ( boldsymbol{y}=boldsymbol{p}(boldsymbol{x}), boldsymbol{y}=boldsymbol{q}(boldsymbol{x}) ) and the
ordinates ( boldsymbol{x}=boldsymbol{a} ) and ( boldsymbol{x}=boldsymbol{b} ) is given by
This question has multiple correct options
A ( cdot int_{a}^{b}(f(x)-g(x)) d x )
B ( cdot int_{a}^{b}(p(x)-q(x)) d x )
c ( cdot int_{a}^{b}|p(x)-q(x)| d x )
D ( cdot int_{a}^{b}|f(x)-g(x)| d x )
12
215 Draw the graph of the linear equations
( 4 x-3 y+4=0 ) and ( 4 x+3 y-20=0 )
Find the area bounded by these lines and x-axis.
12
216 The area bounded by the curves ( y= )
( log x, y=log |x|, y=|log x| ) and ( y= )
( |log | x|| )
A. 4 sq. units
B. 6 sq. units
c. 10 sq. units
D. None of these
12
217 The area of the region(s) enclosed by the curves ( y=x^{2} ) and ( y=sqrt{|x|} ) is:
( A cdot 1 / 3 )
B. 2/3
( c cdot 1 / 6 )
D.
12
218 The area bounded by the curves ( boldsymbol{y}= )
( |x|-1 ) and ( y=-|x|+1 ) is
( mathbf{A} cdot mathbf{1} )
B . 2
c. ( 2 sqrt{2} )
D.
12
219 Area bounded by ( x^{2}=4 a y ) and ( y= )
( frac{8 a^{3}}{x^{2}+4 a^{2}} ) is:
A ( cdot frac{a^{2}}{3}(6 pi-4) )
в. ( frac{pi a^{2}}{3} )
c. ( frac{a^{2}}{3}(6 pi+4) )
D.
12
220 28.
In what ratio does the x-axis divide the area of the region
bounded by the parabolas y = 4x – x2 and y = x2 – x?
(1994 – 5 Marks)
12
221 A polynomial ( P ) is positive for ( x>0 ) and
the area of the region bounded by ( boldsymbol{P}(boldsymbol{x}) )
the ( x- ) axis and the vertical lines ( x=0 )
and ( x=lambda ) is ( frac{lambda^{2}(lambda+3)}{3} ) squnit. Then
polynomial ( boldsymbol{P}(boldsymbol{x}) ) is:
A ( cdot x^{2}+2 x )
B. ( x^{2}+2 x+1 )
c. ( x^{2}+x+1 )
D. ( x^{3}+2 x^{2}+2 )
12
222 If area bounded by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{frac{1}{3}}(boldsymbol{x}-mathbf{1}) )
( x- ) axis is ( A ) then find the value of ( 28 A )
A . 5
B. 6
( c cdot 7 )
D. 9
12
223 Area bounded by the curve ( y^{2}(2 a- )
( x)=x^{3} ) and the line ( x=2 a ) is:
( A cdot 3 pi a^{2} )
B. ( 2 pi )
c. ( 2 pi a )
D. ( 3 pi )
12
224 Find the common area (in sq. units) enclosed by the parabolas ( 4 y^{2}=9 x ) and
( 3 x^{2}=16 y )
12
225 The area bounded by ( y=x^{2} ) and ( y= )
( 1-x^{2} ) is
A ( cdot frac{sqrt{8}}{3} )
в. ( frac{16}{3} )
c. ( frac{32}{3} )
D. ( frac{17}{3} )
12
226 Area bounded by the curve ( mathbf{y}=mathbf{x}+sin mathbf{x} )
and its inverse function between the
ordinates ( mathbf{x}=mathbf{0} ) and ( mathbf{x}=mathbf{2} pi ) is
A. ( 8 pi ) sqp. units
B. 4 ( pi ) sq. units
c. 8 sq. units
D. 3 ( pi ) sq. units
12
227 From a piece of cardboard, in the shape
of a trapezium ( A B C D, ) and ( A B | C D ) and ( angle )
( mathrm{BCD}=90^{circ}, ) quarter circle is removed
Given ( A B=B C=3.5 mathrm{cm} ) and ( mathrm{DE}=2 mathrm{cm} )
Calculate the area of the remaining
piece of the cardboard.(Take ( pi ) to be ( frac{22}{7} )
A ( .9 .625 mathrm{cm}^{3} )
в. ( 6.125 . ) ст
( mathbf{c} cdot 2.625 mathrm{cm}^{2} )
D. None of these
12
228 Let f(x)=Maximum {x’, (1-x), 2x(1 – x)}, where 0 <x< 1.
Determine the area of the region bounded by the curves
y=f(x), X-axis, x= 0 and x=1.
(1997- 5 Marks)
12
229 Find the area of the region bounded by
the curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2}-boldsymbol{x} ) and ( boldsymbol{y}=mathbf{1} )
12
230 Find the area enclosed between the curves ( boldsymbol{y}^{2}-mathbf{2} boldsymbol{y} e^{s i n^{-1} boldsymbol{x}}+boldsymbol{x}^{2}-mathbf{1}+[boldsymbol{x}]+ )
( e^{2 sin ^{-1} x}=0 )
and line ( x=0 ) and ( x=frac{1}{2} ) is (where [.]
denotes greatest integer function)
A. ( frac{sqrt{3}}{4}+frac{pi}{6} )
B. ( frac{sqrt{3}}{2}+frac{pi}{6} )
c. ( frac{sqrt{3}}{4}-frac{pi}{6} )
D. ( frac{sqrt{3}}{2}-frac{pi}{6} )
12
231 The equation of the curve through the point (3,2) and whose slope is ( frac{x^{2}}{y+1}, ) is
( ^{mathrm{A}} cdot frac{y^{2}}{2}+y=frac{x^{3}}{3}+5 )
B . ( y+y^{2}=x^{3}-21 )
c. ( y^{2}+2 y=frac{2 x^{3}}{3}-10 )
D. ( frac{y^{2}}{2}+y=frac{x^{3}}{3}-5 )
12
232 Area of the region bounded by ( y= )
( e^{x}, y=e^{-x}, x=0 ) and ( x=1 ) in sq. units
is:
( ^{A} cdotleft(e+frac{1}{e}right)^{2} )
( ^{text {B }}left(e-frac{1}{e}right)^{2} )
( ^{c} cdotleft(sqrt{e}+frac{1}{sqrt{e}}right)^{2} )
D ( left(sqrt{e}-frac{1}{sqrt{e}}right)^{2} )
12
233 42. The area of the region {(x, y: xy S8,15 y 5x?} is
(JEE Adv. 2018)
(a) Blog 2-14
(b) 16log. 2-14
(c) Blog 2-5
(d) 16log, 2-6
12
234 The area of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse, ( frac{x^{2}}{9}+frac{y^{2}}{5}=1 )
is
A ( cdot frac{27}{4} )
B. 18
c. ( frac{27}{2} )
D. 27
12
235 21.
Sketch the curves and identify the region bounded by
x=
x=2, y=In x and y=2*. Find the area of this region
12
236 Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=boldsymbol{x}, boldsymbol{y}=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) 12
237 46. The area (in sq. units) of the region
{(x, y): x20,x+y<3, x2 < 4y and y < 1+ 1x } is :
[JEEM 2017
59
12
238 Consider the line ( x=sqrt{3} y ) and the
circle ( x^{2}+y^{2}=4 )
What is the area of the region in the first quadrant enclosed by the ( x ) -axis, the line ( x=sqrt{3} y ) and the circle?
A ( cdot frac{pi}{3} )
в.
( ^{C} cdot frac{pi}{3}-frac{sqrt{3}}{2} )
D. None of the above
12
239 The area (in sq. units) of the region
described by ( boldsymbol{A}= )
( left{(x, y) mid y geq x^{2}-5 x+4, x+y geq 1, y leqright. )
is:
A ( cdot frac{17}{6} )
в. ( frac{13}{6} )
c. ( frac{19}{6} )
D. ( frac{7}{2} )
12
240 Find the area of the region bounded by
the curves
( x^{2}+y^{2}=36 ) and ( y^{2}=9 x )
12
241 The area of the figure bounded by ( y^{2}= )
( 2 x+1 ) and ( x-y-1=0 ) is:
( A cdot 2 / 3 )
B . ( 4 / 3 )
( c cdot 8 / 3 )
D. ( 11 / 3 )
12
242 20.
Compute the area of the region bounded by the curves
In x
y=ex In x and y= — where In e=1. (1990 – 4 Mark
ex
12
243 The area bounded by ( y=x e^{|x|} ) and lines
( |boldsymbol{x}|=mathbf{1}, boldsymbol{y}=mathbf{0} ) is
A. 4 sq units
B. 6 sq units
c. 1 sq units
D. 2 sq units
12
244 An open tank with a square base and
vertical side is to be constructed a
metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of
the tank is half of its width.
12
245 The curve ( y=frac{x^{2}}{2} ) and the line ( y=frac{x}{2} )
intersect at the origin and at the point
( (a, b), ) as shown in the figure above. Find
the value of ( b )
( A )
( overline{8} )
B. ( frac{1}{4} )
( c cdot frac{1}{2} )
D.
( E )
12
246 An open box with a square base is to be
made out of a given quantity of cardboard of area ( c^{2} ) square units. Show
that the maximum volume of the box is
( frac{c^{3}}{6 sqrt{3}} ) cubic units.
12
247 17.
The area bounded by the curves y=[xl-1 and y=-x+13
(20025
(a) 1
(6) 2
(c) 2/2
(d) 4
12
248 The tangent to the curve ( y=x^{2}+6 ) at ( a )
point (1,7) touches the circle ( x^{2}+ ) ( y^{2}+16 x+12 y+c=0 ) at a point ( Q )
then the coordinate of ( Q ) then the
coordinate of ( Q ) are.
A. (-6,-11)
B. (-9,-13)
begin{tabular}{l}
c. (-10,-15) \
hline
end{tabular}
D. (-6,-7)
12
249 Find the area of the closed figure bounded by the following curve
( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{2}, boldsymbol{y}=boldsymbol{2}+boldsymbol{4} boldsymbol{x}-boldsymbol{x}^{2} )
12
250 Consider two curves ( C_{1}: y=frac{1}{x} ) and ( C_{2} )
( boldsymbol{y}=ln x ) on the xy plane Let ( boldsymbol{D}_{1} ) denotes
the region surrounded by ( C_{1}, C_{2} ) and the
line ( x=1 ) and ( D_{2} ) denotes the region
surrounded by ( C_{1}, C_{2} ) and the line ( x=a )
If ( D_{1}=D_{2} ) then the value of ‘a’:
A ( cdot frac{e}{2} )
в.
( mathbf{c} cdot e-1 )
D. ( 2(e-1) )
12
251 The area bounded by two branches of the curve ( (boldsymbol{y}-boldsymbol{x})^{2}=boldsymbol{x}^{3} & boldsymbol{x}=mathbf{1} ) equals
A . ( 3 / 5 )
B. ( 5 / 4 )
( c cdot 6 / 5 )
D. ( 4 / 5 )
12
252 The area between the curve ( y^{2}=9 x ) and
the line ( y=3 x ) is
A ( cdot frac{1}{3} ) sq. units
B. ( frac{8}{3} ) sq. units
c. ( frac{1}{2} ) sq, units
D. ( frac{1}{5} ) sq. units
12
253 1.
Find the area bounded by the curve xú= 4y
line x=4y-2.
e area bounded by the curve x2=4y and the straight
(1981 – 4 Marks)
12
254 Find the area of the region bounded by
the parabola ( y^{2}=2 x ) and the line ( x- )
( boldsymbol{y}=mathbf{4} . ) And ( mathbf{1 8} boldsymbol{s} boldsymbol{q} ) unit.
12
255 The number of solutions for
( sin left(frac{pi x}{2}right)=frac{99 x}{500} ) is:
( A cdot 3 )
B. 5
( c cdot 7 )
( D )
12
256 If ( A ) is the area of the figure bounded by
the straight lines ( x=0 ) and ( x=2, ) and
the curves ( y=2^{x} ) and ( y=2 x-x^{2} ) then the value of ( 672left(frac{3}{log 2}-Aright) ) is
12
257 Area of the region bounded by ( boldsymbol{y}=|boldsymbol{x}| )
and ( boldsymbol{y}=mathbf{1}-|boldsymbol{x}| ) is
A ( cdot frac{1}{3} ) sq. units
B. 1 sq. units
c. ( frac{1}{2} ) sq. unit
D. 2 sq. units
12
258 A curve passes through the point ( (2 a, a) ) and is such that sum of subtangent and abscissa is equal to a. Its equation is
A ( cdot(x-a) y^{2}=a^{3} )
в. ( (x-a)^{2} y=a^{3} )
c. ( (x-a) y=a^{2} )
D. ( (x+a) y=a^{2} )
12
259 The area bounded by curves ( 3 x^{2}+ )
( mathbf{5} boldsymbol{y}=mathbf{3} 2 ) and ( boldsymbol{y}=|boldsymbol{x}-mathbf{2}| ) is
A . 25
B. 17/2
c. ( 33 / 2 )
D. 33
12
260 Set up an equation of a tangent to the graph of the following function.

A sector with a central angle ( alpha ) is cut off
from a circle. A cone is made of the
remaining part of the circle. At what
value of ( alpha ) is the capacity of the cone
the greatest?

12
261 The area bounded by ( |y|=1-x^{2} ) is
( A cdot 8 / 3 )
в. ( 4 / 3 )
( mathrm{c} cdot 16 / 3 )
D. None of these
12
262 Tangents are drawn to the ellipse ( frac{x^{2}}{9}+ ) ( frac{y^{2}}{5}=1 ) at the ends of both latus
rectum. The area of the quadrilateral so formed is
A. 27 sq.units
B. ( frac{13}{2} ) sq.units
c. ( frac{15}{4} ) sq.units
D. 45 sq.units
12
263 If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})-boldsymbol{x} boldsymbol{y} ) for all
( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( lim _{boldsymbol{h} rightarrow mathbf{0}} frac{boldsymbol{f}(boldsymbol{h})}{boldsymbol{h}}=mathbf{3}, ) then the
area bounded by the curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) )
and ( boldsymbol{y}=boldsymbol{x}^{2} ) is:
A . 1
B. 2
( c cdot 3 )
D. 4
12
264 The area of the region
( left{(x, y): x y leq 8,1 leq y leq x^{2}right} ) is
( mathbf{A} cdot 16 log _{6} 2-6 )
в. ( 8 log _{6} 2-frac{7}{3} )
( mathrm{c} cdot_{16 log _{6} 2-frac{14}{3}} )
D. ( 8 log _{6} 2-frac{14}{3} )
12
265 38. Let f(x) be a continuous function given by
(x) = { 2x
lxs11
(1999 – 10 Marks)
(x2 + ax + b, | x | >1)
Find the area of the region in the third quadrant bounded by
the curves x=-2y2 and y=f(x) lying on the left of the line
8x+1=0.
12
266 The function ( f(x) ) whose graph passes
through the point ( left(0, frac{7}{3}right) ) and whose derivatives is ( x sqrt{1-x^{2}} ) is given by
12
267 The area (in square units) bounded by
the curves ( x=-2 y^{2} ) and ( x=1-3 y^{2} )
is
A ( cdot frac{2}{3} )
B.
( c cdot frac{4}{3} )
D.
12
268 The area(in sq. units) of the smaller portion enclosed between the curves, ( x^{2}+y^{2}=4 ) and ( y^{2}=3 x, ) is
A ( cdot frac{1}{sqrt{3}}+frac{4 pi}{3} )
в. ( frac{1}{2 sqrt{3}}+frac{pi}{3} )
c. ( frac{1}{2 sqrt{3}}+frac{2 pi}{3} )
D. ( frac{1}{sqrt{3}}+frac{2 pi}{3} )
12
269 Let ( f(x)=x^{2}-3 x+2 ) then area
bounded by the curve ( f(|x|) ) (in square
units) and ( x ) -axis is
A ( cdot frac{1}{3} )
в. ( frac{5}{6} )
c. ( frac{5}{3} )
D. None of thes
12
270 The area bounded by the ( y ) -axis, ( y= ) ( cos x ) and ( y=sin x ) when ( 0 leq x leq frac{pi}{2} )
A ( cdot 2(sqrt{2}-1) )
B. ( sqrt{2}-1 )
( c cdot sqrt{2}+1 )
D. ( sqrt{2} )
12
271 State the following statement is True or False
The area bounded by the circle ( x^{2}+ ) ( boldsymbol{y}^{2}=mathbf{1}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{4} ) and the pair of lines
( sqrt{3}left(x^{2}+y^{2}right)=4 x y, ) is equal to ( frac{pi}{2} . ) The
statement is true or false.
A. True
B. False
12
272 Assertion
If ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2})(boldsymbol{x}-mathbf{3}), ) then
area enclosed by ( |boldsymbol{f}(boldsymbol{x})| ) between the
lines ( x=2.2, x=2.8 ) and ( x- ) axis is equal to ( int_{2.2}^{2.8}(x-1)(x-2)(x-3) d x )
Reason
( (x-1)(x-2)(x-3) leq 0 ) for all ( x in )
[2.2,2.8]
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
273 The positive value of ( k ) for which ( mathbf{k e}^{mathbf{x}}- )
( mathbf{x}=mathbf{0} ) has only one root is
A . ( 1 / e )
B. 1
( c )
D. ( log _{mathrm{e}} 2 )
12
274 The area bounded by ( y=f(x), x-a x i s )
and the line ( y=1, ) where ( f(x)=1+ ) ( frac{1}{x} int_{1}^{x} f(t) d t ) is
A ( cdot 2(e+1) )
B ( cdotleft(1-frac{1}{e}right) )
c. ( 2(e-1) )
D. None of these
12
275 ( A: ) The area bounded by ( x=3, y^{2}=3 x )
( B: ) The area bounded by ( y=1-|x| ) and
X-axis
( C: ) The area enclosed between the curve
( y=x^{2} ) and the line ( y=sqrt{3} x )
The descending order of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) is
A. ( A, C, B )
в. ( C, B, A )
c. ( A, B, C )
D. ( C, A, B )
12
276 Find the area of that part of the circle
( x^{2}+y^{2}=16 ) which is exterior to the
parabola ( boldsymbol{y}^{2}=mathbf{6} boldsymbol{x} )
12
277 The common area between the curve
( x^{2}+y^{2}=8 ) and ( y^{2}=2 x ) is
A ( cdot frac{4}{3}+2 pi )
B. ( (2 sqrt{2}+pi-1) )
c. ( (sqrt{2}+pi-1) )
D. None of these
12
278 The area bounded by the curves ( y=x^{2} )
and line ( boldsymbol{y}=boldsymbol{x} )
( A cdot sin 1 )
B. ( 1-sin 1 )
c. ( 1+sin 1 )
D. None of these
12
279 The area of the region bounded by the curves ( boldsymbol{y}=boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{y}^{2} ) is
A ( cdot frac{1}{3} )
в. ( frac{1}{2} )
( c cdot frac{1}{4} )
D. 3
12
280 For a quadratic function in standard
form, ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c}, ) Find the axis of
symmetry.
A ( cdot y=frac{-b}{2 a} )
в. ( x=frac{-b}{2 a} )
c. ( y=frac{b^{2}-4 a c}{2 a} )
D. None of these
12
281 Find the area of the closed figure bounded by the following curves ( boldsymbol{y}=sqrt{boldsymbol{x}}, boldsymbol{y}=sqrt{mathbf{4}-mathbf{3} boldsymbol{x}}, mathbf{y}=mathbf{0} )
A ( cdot frac{8}{9} )
B. ( frac{7}{9} )
( c cdot frac{5}{9} )
D.
12
282 Find the point(s) on the curve ( y^{3}+ )
( 3 x^{2}=12 y ) where the tangent is vertical
(parallel to ( y-a x i s) )
12
283 Find the area enclosed between the
parabola ( 4 y=3 x^{2} ) and the straight line
( 3 x-2 y+12=0 )
12
284 Find the area of one of the curvilinear
triangles formed by ( boldsymbol{y}=sin boldsymbol{x}, boldsymbol{y}= )
( cos x ) and ( x ) axis
12
285 The area bounded by the parabola ( boldsymbol{x}= )
( y^{2} ) and the line ( y=x-6 ) is
A ( cdot frac{125}{3} ) sq. units
B. ( frac{125}{6} ) sq. units
c. ( frac{125}{4} ) sq. units
D. ( frac{115}{3} ) sq. units
12
286 Find the point on the curve ( y=x^{2} )
where rate of change of ( x ) -co ordinate is
equal to the rate of change of ( y-c o )
ordinate.
12
287 If the area enclosed by the curves ( y^{2}= )
( 4 lambda x ) and ( y=lambda x ) is ( frac{1}{9} ) square units then
value of ( lambda ) is
A .24
B. 37
c. 48
D. 38
12
288 A farmer ( F_{1} ) has a land in the shape of a
triangle with vertices at ( boldsymbol{P}(mathbf{0}, mathbf{0}), boldsymbol{Q}(mathbf{1}, mathbf{1}) )
and ( R(2,0) . ) From this land, a
neighbouring farmer ( F_{2} ) takes away the region which lies between the side PQ and a curve of the form ( boldsymbol{y}=boldsymbol{x}^{n}(boldsymbol{n}>1) )
If the area of the region taken away by the farmer ( F_{2} ) is exactly ( 30 % ) of the area of ( Delta P Q R, ) then the value of ( n ) is
12
289 The area in square units bounded by the
curves ( boldsymbol{y}=boldsymbol{x}^{3}, boldsymbol{y}=boldsymbol{x}^{2} ) and the
ordinates ( boldsymbol{x}=mathbf{1}, boldsymbol{x}=mathbf{2} ) is
A ( cdot frac{17}{12} )
B. ( frac{12}{13} )
( c cdot frac{2}{7} )
( D cdot frac{7}{2} )
12
290 The area bounded by curve ( y= )
( sin 2 x(x=0 quad text { to } quad x=pi) ) and ( X ) -axis is
A .4
B. 2
c.
D.
12
291 The area (in sq. units) of the region
bounded by the curves ( y=2^{x} ) and ( y= )
( |x+1|, ) in the first quadrant is:
( ^{mathbf{A}} cdot frac{3}{2}-frac{1}{log _{e} 2} )
в. ( frac{1}{2} )
( ^{mathrm{C}} cdot log _{e} 2+frac{3}{2} )
D.
12
292 The area bounded by the parabola ( y^{2}= )
( 4 x ) and its latusrectum is:
A ( cdot frac{8}{3} ) sq. units
B. ( frac{3}{8} ) sq. units
c. 12 sq. units
D. ( frac{1}{3} ) sq. units
12
293 The radius of a circular garden is ( 90 mathrm{m} ) There is a road ( 9 mathrm{m} ) wide around it. Find
total area of the road and the cost of
levelling the road at Rs.5 per sq.m.
12
294 If the curve ( x+y=x^{2}(x+1) ) has two
distinct horizontal tangents, then the distance between them is
A ( cdot frac{32}{27} )
в. ( frac{27}{32} )
( c cdot frac{4}{3} )
D. ( frac{22}{27} )
12
295 If ( 0<A<frac{pi}{6} ) then ( A(csc A) ) is:
( A cdotfrac{pi}{3} )
( c cdot=frac{pi}{3} )
D. ( =frac{pi}{6} )
12
296 The area common to the circle ( x^{2}+ )
( y^{2}=16 a^{2} ) and the parabola ( y^{2}=6 a x ) is
A ( cdot 4 a^{2}(8 pi-sqrt{3}) )
B. ( frac{4 a^{2}(4 pi+sqrt{3})}{3} )
c. ( frac{8 a^{2}(4 pi-sqrt{3})}{5} )
D. none of these
12
297 The line ( 2 y=3 x+12 ) cuts the
parabola ( 4 y=3 x^{2} . ) What is the area
enclosed by the parabola and the line?
A. 27 square unit
B. 36 square unit
c. 48 square unit
D. 54 square unit
12
298 What is the greatest integer value of ( x ) for which ( 2 x-20<0 ? )
( A cdot 8 )
B.
c. 10
D. 1
12
299 Sketch for ( boldsymbol{y}=sin ^{-1}left(frac{1+x^{2}}{2 x}right) ) 12
300 The area enclosed between the curves
( boldsymbol{y}=boldsymbol{x}^{3} ) and ( boldsymbol{y}=sqrt{boldsymbol{x}} ) is, (in square
units):
A ( cdot frac{5}{3} )
в.
c. ( frac{5}{12} )
D. ( frac{12}{5} )
12
301 Area enclosed between the curves ( |boldsymbol{y}|= )
( 1-x^{2} ) and ( x^{2}+y^{2}=1 ) is
A ( cdot frac{3 pi-14}{3} ) sq.units
B. ( frac{pi-8}{3} ) sq.units
c. ( frac{2 pi-8}{3} ) sq.units
D. None of these
12
302 26. The area enclosed between the curves y = ax? and
x=ay2 (a > 0) is 1 sq. unit, then the value of a is (2004S)
(a) 1/13 (b) 1/2
(c) 1
(d) 1/3
12
303 The sum of the intercepts of the tangent to the curve ( sqrt{x}+sqrt{y}=3 ) on the coordinate axes where it meets the curve
( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} )
( A cdot S )
B. 10
( c cdot 11 )
D. 12
12
304 Radius of the circle that passes through
origin and touches the parabola ( boldsymbol{y}^{2}= )
( 4 a x ) at the point ( (a, 2 a) ) is
A ( cdot frac{5}{sqrt{2}} a )
В. ( 2 sqrt{2} a )
c. ( sqrt{frac{5}{2} a} )
D. ( frac{3}{sqrt{2}} a )
12
305 Find the area under the curve ( boldsymbol{y}= )
( left(x^{2}+2right)^{2}+2 x ) between the lines ( x=0 )
( x=2 ) and the ( X ) -axis
12
306 If area bounded by to curves ( boldsymbol{y}^{2}=4 a x )
and ( y=m x ) is ( frac{a^{2}}{3}, ) then the value of ( m ) is
A .2
B. – 1
( c cdot frac{1}{2} )
D. none of these
12
307 (2005 – 2 Marks)
46. Find the area bounded by the curves x2 = y, x = -y and
p=4x-3.
(2005 AM
the curves 2005 22 Mark and
12
308 Find the area of the circle ( 4 x^{2}+4 y^{2}= )
9 which is interior to the parabola ( x^{2}= )
( 4 y )
12
309 Find the area of the sector of a circle
bounded by the circle ( x^{2}+y^{2}=16 ) and
the line ( y=x ) in the first quadrant.
12
310 For a real number ( x ) let ( [x] ) denote the
largest number less than or equal to ( x ) for ( boldsymbol{x} epsilon boldsymbol{R} ) let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] sin pi boldsymbol{x} . ) Then
A. fis differentiable on R
B. fis symmetric about the line ( x=0 )
( ^{mathbf{c}} cdot int_{-3}^{3} f(x) d x=0 )
D. For each real ( alpha ), the equation ( f(x)-alpha=0 ) has infinitely many roots.
12
311 Area enclosed between the curves ( y^{2}= )
( x ) and ( x^{2}=y ) is equal to
This question has multiple correct options
A ( cdot_{2} int_{0}^{1}left(x-x^{2}right) d x )
B.
c. area of region ( left{(x, y): x^{2} leq y leq|x|right} )
D. ( frac{2}{3} )
12
312 Area enclosed by the graph of the function ( y=ln ^{2} x-1 ) lying in the ( 4 t h )
quadrant is
( A cdot frac{2}{e} )
B. ( frac{4}{e} )
c. ( _{2}left(e+frac{1}{e}right) )
D. ( 4left(e-frac{1}{e}right) )
12
313 Area of region ( left{(x, y) in R^{2}: y geq sqrt{|x+3|}, 5 y leq x+9right. )
is equal to
A ( cdot frac{1}{6} )
B. ( frac{4}{3} )
( c cdot frac{3}{2} )
D.
12
314 Find the value(s) of the parameter’a’ (a
9) for each of which the area of the
figure bounded by the straight line ( y= ) ( frac{a^{2}-a x}{1+a^{4}} & ) the parabola ( y= )
( frac{x^{2}+2 a x+3 a^{2}}{1+a^{4}} ) is the greatest
A ( cdot a=2^{1 / 4} )
В . ( a=5^{1 / 4} )
c. ( a=7^{1 / 4} )
12
315 19.
The parabolas y2 = 4x and x2 = 4y divide the square
region bounded by the lines x = 4, y = 4 and the coordinate
axes. If S1, S2, Sz are respectively the areas of these parts
numbered from top to bottom; then S: S2: S3 is [2005]
(a) 1:2:1 (b) 1:2:3 (c) 2:1:2 (d) 1:1:1
12
316 Draw the curve represented by ( sqrt{x}+ ) ( sqrt{boldsymbol{y}}=mathbf{1} ) 12
317 The area enclosed between the curves
( boldsymbol{y}=boldsymbol{x}^{3} ) and ( boldsymbol{y}=sqrt{boldsymbol{x}} ) is (in square units)
A ( cdot frac{5}{3} )
B. ( frac{5}{4} )
c. ( frac{5}{12} )
D. ( frac{12}{5} )
12
318 16.
If the line sx = a divides the area of region
R = {(x,y) = R2 : x’ sysx,0sx51} into two equal
parts, then
(JEE Adv. 2017)
(a) Ocasi (b) f<a<i
(c) 2a4 -4a² +1=0 (d) a4 + 4a2 -1 = 0
12
319 The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= )
1 is equal to
A ( cdot frac{5}{3}^{operatorname{sq} . u n i t} )
в. ( frac{1}{3} ) sq.unit
c. ( frac{2}{3} ) squnit
D ( cdot frac{4}{3} ) sq.unit
12
320 Find the area bounded between the
curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=sqrt{boldsymbol{x}} )
12
321 The area bounded by the curve ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} )
and the lines ( boldsymbol{y}=|boldsymbol{x}-mathbf{1}|, boldsymbol{x}=mathbf{2} ) is
given by
( mathbf{A} cdot e^{2}+1 )
B ( cdot e^{2}-1 )
c. ( e^{2}-2 )
D. ( e-2 )
12
322 Find the area of the shaded region in
fig:- if ( A B C D ) is a square of side ( 14 mathrm{cm} ) and ( A P D times B P C ) are semi circles.
12
323 Using the method of integration find the area bounded by the curve ( |boldsymbol{x}|+|boldsymbol{y}|=mathbf{1} ) 12
324 Area laying between the curves ( y= )
( tan x, y=cot x ) and ( x- )
( boldsymbol{a x i s}, boldsymbol{x}left[boldsymbol{o}, frac{pi}{2}right] boldsymbol{i s} )
12
325 The closest distance of the origin from a curve given as ( boldsymbol{a} overline{boldsymbol{z}}+overline{boldsymbol{a}} boldsymbol{z}+boldsymbol{a} overline{boldsymbol{a}}=mathbf{0}(boldsymbol{a} ) is a
complex number
A . 1
в. ( frac{|a|}{2} )
c. ( frac{operatorname{Re}(a)}{|a|} )
D. ( frac{operatorname{Im}(a)}{|a|} )
12
326 The area enclosed by the curves ( x^{2}= )
( y, y=x+2 ) and ( x ) -axis is:
A ( cdot frac{5}{6} )
B. ( frac{5}{4} )
( c cdot frac{5}{2} )
D.
12
327 Find the area of the shaded portion in
the given figure, where ( A B C D ) is a
square of side ( 14 mathrm{cm} ) and semicircles
are drawn with each side of square as
diameter.
12
328 33. Letf:/-1,2] → [0, 0 ) be a continuous function such that
f(x)=f(1-x) for all x € (-1, 2]
Let R1 =
xf (x)dx , and R, be the area of the region
-1
(2011)
bounded by y=f(x),x=-1, x=2, and the x-axis.
Then
(a) R1 = 2R2
(b) R1 = 3R
© 2R1 = R2
(d) 3R = R2
12
329 The area bounded by ( y=sec ^{-1} x, y= )
( operatorname{cosec}^{-1} x ) and the line ( x-1=0 ) is:
( mathbf{A} cdot ln (3+2 sqrt{2})-frac{pi}{2} )
В ( cdot frac{pi}{2}+ln (3+2 sqrt{2}) )
( c cdot pi-ln 3 )
D . ( pi+ln 3 )
12
330 A polynomial function ( f(x) ) satisfies the condition ( boldsymbol{f}(boldsymbol{x}+mathbf{1})=boldsymbol{f}(boldsymbol{x})+mathbf{2} boldsymbol{x}+mathbf{1} )
Find ( boldsymbol{f}(boldsymbol{x}) ) if ( boldsymbol{f}(mathbf{0})=1 . ) Find also the
equations of the pair of tangents from
the origin on the curve ( y=f(x) ) and
compute the area enclosed by the curve
and the pair of tangents.
( f(x)=x^{2}+1 ; y=pm 2 x ;, A=frac{2}{3} ) sq.units
B. ( f(x)=x^{2}-1 ; y=pm 2 x ; A=frac{2}{3} ) sq.units
c. ( f(x)=x^{2}+1 ; y=pm 2 x ; ), ( A=frac{3}{2} ) sq.units
D. ( f(x)=x^{2}-1 ; y=pm 2 x ; A=frac{3}{2} ) sq.units
12
331 Find the area bounded by the curve ( y= ) ( sin x, 0 leq x leq pi ) and line ( y=frac{1}{sqrt{2}} ) 12
332 f ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} forall boldsymbol{x} in[mathbf{0}, boldsymbol{x} / 2], boldsymbol{f}(boldsymbol{x})+ )
( boldsymbol{f}(boldsymbol{pi}-boldsymbol{x})=boldsymbol{2} forall boldsymbol{x} in[boldsymbol{x} / 2, boldsymbol{pi}] ) and
( boldsymbol{f}(boldsymbol{x})=boldsymbol{f}(boldsymbol{2} boldsymbol{pi}-boldsymbol{x}) forall boldsymbol{x} in[boldsymbol{pi}, boldsymbol{2} boldsymbol{pi}] ) then
find the area bounded by ( y=f(x) ) and
the ( x- ) axis.
12
333 The area of the region bounded by the
curves ( boldsymbol{y}=mathbf{2}^{x}, boldsymbol{y}=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) and ( boldsymbol{x}=mathbf{2} )
is
A ( cdot frac{3}{log 2}-frac{4}{3} )
в. ( frac{3}{log 2}-frac{4}{9} )
c. ( frac{3}{2}-frac{log 2}{9} )
D. None of these
12
334 36. The area of the region enclosed by the curves
1201
y=x, x = e, y = – and the positive x-axis is
square unit
square units
square units
square unit
12
335 A stone is dropped into a quiet lake and waves move in circles at the sped of 5
cm/sec.

At that instant, when radius of circular
wave is ( 8 mathrm{cm}, ) how fast is the enclosed area increasing?
( mathbf{A} cdot 8 pi mathrm{cm}^{2} / mathrm{s} )
B. ( 80 pi c m^{2} / s )
c. ( 6 pi c m^{2} / s )
D. ( frac{8}{3} pi c m^{2} / s )

12
336 A point P moves in xy-plane In such a way that ( [|x|]+[|y|]=1 ) were [.] denotes the greatest integer function. Area of the region representing all possible positions of the point ‘P’ is equal to
A. 4 sq. units
B. 16 sq. units
c. ( 2 sqrt{2} ) sq. units
D. 8 sq. units
12
337 Find the area of the region bounded by the x-axis and me
curves defined by
(1984 – 4 Marks)
TT
y = tan x,

; y = cot x,
<<3
12
338 If ( A_{1} ) is the area bounded by ( y= )
( cos x, y=sin x & x=0 ) and ( A_{2} ) the
area bounded by ( boldsymbol{y}=cos boldsymbol{x}, boldsymbol{y}= )
( sin x, y=0 ) in ( left(0, frac{pi}{2}right) ) then ( frac{A_{1}}{A_{2}} ) equals to:
A ( cdot frac{1}{2} )
B. ( frac{1}{sqrt{2}} )
c.
D. None of these
12
339 In the diagram ( square A B C D ) is the
rectangular paper. If ( A B=20 mathrm{cm} ) and
( B C=14 mathrm{cm} ) then what is the area of
the shaded region if the semicircle
with diameter ( B C ) is cut from the
paper.
12
340 The area under the curve ( y=2 sqrt{x} )
bounded by the lines ( x=0 ) and ( x=1 ) is
( A cdot frac{4}{3} )
B. ( frac{2}{3} )
c. 1
D.
12
341 Find the area of the closed figure bounded by the following curve ( y=x^{2}, y=2 x-x^{2} ) 12
342 Area of the region bounded by rays ( |x|+y=1 ) and ( X ) -axis is
A ( cdot frac{1}{2} )
B. 2
c.
D.
12
343 The parabolas ( y^{2}=4 x ) and ( x^{2}=4 y )
divide the square region bounded by the lines ( x=4, y=4 ) and the coordinate axes.
f ( S_{1}, S_{2}, S_{3} ) are respectively the areas of
these parts numbered from top to
bottom(Example: ( S_{1} ) is the area
bounded by ( y=4 ) and ( x^{2}=4 y ) ); then
( boldsymbol{S}_{1}, boldsymbol{S}_{2}, boldsymbol{S}_{3} ) is
A . 1: 2: 1
B. 1: 2: 3
c. 2: 1: 2
D. 1: 1: 1
12
344 The area included between the
parabolas ( y^{2}=4 x ) and ( x^{2}=4 y ) is
A ( cdot frac{8}{3} ) sq unit
B. 8 sq unit
c. ( frac{16}{3} ) sq unit
D. 12 sq unit
12
345 The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= )
( mathbf{1} ) is
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
( c cdot frac{4}{3} )
D.
12
346 Consider an ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) What is the area included between the ellipse and the greatest rectangle inscribed in the ellipse?
A ( . a b(pi-1) )
B. ( 2 a b(pi-1) )
c. ( a b(pi-2) )
D. None of the above
12
347 18.
The triangle formed by the tangent to the curve f(x)=x-T DX
at the point (1, 1) and the coordinate axes, lies in the first
quadrant. If its area is 2, then the value of bis (20015)
(a) -1 (b) 3
(c) 3
(d) 1
12
348 Find the area (sq.units) bounded by ( boldsymbol{y}=sin ^{-1} boldsymbol{x} ) and ( boldsymbol{y}=cos ^{-1} boldsymbol{x} ) and ( mathbf{x} ) -axis. 12
349 Find area curved by three circles
( mathbf{A} cdot(5 pi-3 sqrt{3}) ) units ( ^{2} )
( mathbf{B} cdot(5 pi+4 sqrt{3}) ) units ( ^{2} )
( mathbf{c} cdot(5 pi+3 sqrt{3}) ) units ( ^{2} )
( mathbf{D} cdot(5 pi+3 sqrt{2}) ) units ( ^{2} )
12
350 9.
The slope of the tangent to a curve y = f(x) at (x, f(x)] is
2x + 1. If the curve passes through the point (1,2), then the
area bounded by the curve, the x-axis and the line x = 1 is
(1995)
(d)
6
12
351 5. Find the area of the region bounded by the curva
π
C:y=tan x, tangent drawn to Catx= ã and the x-axis.
12
352 A hemispherical tank of radius ( boldsymbol{R} ) is
completely filled with water. Now an orifice of small area ‘a’ is made at the
bottom of tank. The time required to
empty the tank is
( ^{mathrm{A}} cdot frac{14 pi R^{frac{5}{2}}}{15 sqrt{2} a sqrt{g}} )
( ^{mathrm{B}} cdot frac{14 pi R^{frac{3}{2}}}{sqrt{2} a sqrt{g}} )
c. ( frac{14 pi R^{frac{5}{2}}}{5 sqrt{2} a sqrt{g}} )
( frac{4 pi R^{frac{5}{2}}}{15 sqrt{2} a sqrt{g}} )
12
353 The area bounded by the curve ( y= ) ( (x+1)^{2}, y=(x-1)^{2} ) and the line ( y= )
0 is
A ( cdot frac{1}{6} )
в. ( frac{2}{3} )
( c cdot frac{1}{4} )
D.
12
354 The area bounded by ( y^{2}=4 x ) and ( x^{2}= )
( 4 y ) is
A ( -frac{20}{3} ) sq. units
B. ( frac{16}{3} ) sq. units
( ^{mathrm{c}} cdot frac{14}{3} mathrm{sq} . ) units
D. ( frac{10}{3} ) sq. units
12
355 What is the area of the region bounded
by the lines ( x=y, y=0 ) and ( x=4 ? )
A. 4 square units
B. 8 square units
c. 12 square units
D. 16 square units
12
356 The area lying in the first quadrant inside the circle ( x^{2}+y^{2}=12 ) and
bounded by the parabolas ( boldsymbol{y}^{2}= )
( 4 x, x^{2}=4 y ) is:
A ( cdotleft(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) )
B ( cdot 4left(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) )
( ^{c} cdotleft(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) )
D. none of these
12
357 The area (in sq. units) of the region ( A=left{(x, y): x^{2} leq y leq x+2right} ) is?
A ( cdot frac{10}{3} )
B. ( frac{9}{2} )
c. ( frac{31}{6} )
D. ( frac{13}{6} )
12
358 The whole area of the curves ( x= )
( a cos ^{3} t, y=b sin ^{3} t ) is given by?
A ( cdot frac{3}{8} pi a b )
в. ( frac{5}{8} pi a b )
c. ( frac{1}{8} pi a b )
D. None of these
12
359 Show that the right circular cylinder of given surface and maximum volume is
such that its height is equal to the diameter of the base.
12
360 If ( f(x)=max left{sin x, cos x, frac{1}{2}right}, ) then the
area of the region bounded by the
curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{x}- ) axis ( boldsymbol{y}- ) axis and
( boldsymbol{x}=2 pi ) is
( ^{mathbf{A}} cdotleft(frac{5 pi}{12}+3right) cdot ) sq.unit
в. ( left(frac{5 pi}{12}+sqrt{2}right) ). sq.unit
c. ( left(frac{5 pi}{12}+sqrt{3}right) cdot ) sq.unit
D ( cdotleft(frac{5 pi}{12}+sqrt{2}+sqrt{3}right) cdot ) sq.unit
12
361 Find the area between the ( x ) -axis and
the curve ( y=sin x ) from ( x=0 ) to ( x=2 pi )
12
362 For which of the following values of ( m ) the area of the region bounded by the
curve ( y=x-x^{2} ) and the line ( y=m x ) equals ( frac{mathbf{9}}{mathbf{2}} )
A . -4
B. –
c. 2
D. 4
12
363 The area between the curve ( y^{2}=9 x ) and
the line ( y=3 x ) is
A ( cdot frac{1}{3} ) sq units
B. ( frac{8}{3} ) sq. units
c. ( frac{1}{2} ) sq. units
D. ( frac{1}{5} ) sq. units
12
364 The area of the smaller part bounded by the semi-circle ( y=sqrt{4-x^{2}}, y=x sqrt{3} )
and ( x ) -axis is
A ( cdot frac{pi}{3} )
в. ( frac{2 pi}{3} )
c. ( frac{4 pi}{3} )
D. none of these
12
365 Find the area of the region bounded by
the curves ( y=log _{e} x, y=sin ^{4} pi x, x= )
( mathbf{0} )
A ( cdot frac{11}{8} ) sq.units
B. ( frac{9}{8} ) sq.units
c. ( frac{13}{8} ) sq.units
D. ( frac{15}{8} ) sq.units
12
366 Assertion
Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{m} boldsymbol{i} boldsymbol{n} cdot(boldsymbol{x}+1, sqrt{1-boldsymbol{x}}), ) then
area bounded by ( y=f(X) & x ) -axis is
( mathbf{7} )
( overline{6} ) square units.
Reason
( min ) of ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}x+1 & text { for }-1 leq x<0 \ sqrt{1-x} & , 0<x leq 1end{array}right. )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
367 The area enclosed between the curve
( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}^{2}=boldsymbol{y} ) is equal to
12
368 The area bounded by the curve ( y= )
( x^{4}-2 x^{3}+x^{2}+3 ) with ( x ) -axis and
ordinates corresponding to the minima of y is :
( mathbf{A} cdot mathbf{1} )
B . ( frac{91}{30} )
( c cdot frac{30}{9} )
D. 4
12
369 Draw the curve ( y=sin ^{4} x ) between ( x= )
0 and ( x=2 pi )
12
370 ( A B ) and ( C D ) are two perpendicular diameters of a circle with centre ( 0,0 D ) is
the diameter of smaller circle.If ( O A=7 )
( mathrm{cm} ),then the area of the shaded region in
figure,is
( A cdot 66.5 c m^{2} )
В. ( 66 c m^{2} )
( c cdot 65 c m^{2} )
D. ( 67.5 mathrm{cm}^{2} )
12
371 Use the test for symmetry to determine
if the graph of ( y-5 x^{2}=4 ) is
symmetric about the x-axis.
A. Symmetrical about x-axis
B. Not symmetrical about x-axis
c. Can’t be determined
D. None of these
12
372 The area of the figure bounded by two branches of the curve ( (boldsymbol{y}-boldsymbol{x})^{2}=boldsymbol{x}^{3} )
and the straight line ( x=1 ) is:
A ( cdot frac{1}{3} ) sq.unit
B ( cdot frac{4}{5} ) sq.unit
c. ( frac{5}{4} ) sq.unit
D. 3 sq.unit
12
373 7.
[200
The area of the region bounded by the curves
y = x – 1 and y = 3-|xis
(a) 6 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) 4 sq. units.
12
374 The area bounded by the curves ( y= )
( x(x-3)^{2} ) and ( y=x ) is (in sq.units):
A . 28
B. 32
( c cdot 4 )
D. 8
12
375 Tangent is drawn from (1,0) to ( y=e^{x} )
then the area bounded between the
coordinate axes and the tangent is
equal to –
A ( cdot frac{e}{2} )
B. ( e )
( ^{c} cdot frac{e^{2}}{2} )
( mathbf{D} cdot e^{2} )
12
376 Smaller area enclosed by the circle ( x^{2}+y^{2}=4 ) and the line ( x+y=2 ) is
A. ( 2(pi-2) )
B. ( pi-2 )
c. ( 2 pi-1 )
D. ( 2(pi+2) )
12
377 A stone is dropped into a quiet lake an waves move in circles at the speed of 5 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius of the circular wave is ( 8 mathrm{cm}, ) how fast is the enclosed area increasing? 12
378 The area bounded by ( y=3 x ) and ( y=x^{2} ) is (in square units)
A . 10
B. 5
c. 4.5
D.
12
379 Find the area of the region bounded by
the curve ( y=4-x^{2}, x ) -axis and the line
( x=0 ) and ( x=2 )
A ( cdot frac{11}{3} )
в. ( frac{16}{3} )
( c cdot frac{16}{5} )
D.
12
380 Form the differential equation of the family of curves represented by the equation(a being the parameter). ( (x-a)^{2}+2 y^{2}=a^{2} ) 12
381 The area (in square units) bounded by the curves ( boldsymbol{y}=sqrt{boldsymbol{x}} )
( 2 y-x+3=0, ) and lying in the first
quadrant is
12
382 12. If y = x², then area of curve y v/s x from x = 0 to 2 will be:
(a) 1/3 (b) 8/3 (c) 4/3 (d) 2/3
12
383 Rewrite the equation ( |boldsymbol{y}|=|boldsymbol{x}| ) as two
equations of two lines.
12
384 The area of the region lying between the
line ( boldsymbol{x}-boldsymbol{y}+mathbf{2}=mathbf{0} ) and the curve ( boldsymbol{x}= )
( sqrt{boldsymbol{y}} ) is
( mathbf{A} cdot mathbf{9} )
B. ( frac{9}{2} )
c. ( frac{10}{3} )
D. none
12
385 The area of the region bounded by the curves ( boldsymbol{f}(boldsymbol{x})=operatorname{maximum}left{|boldsymbol{x}|, boldsymbol{x}^{2}, sqrt{boldsymbol{x}}right} )
where ( boldsymbol{x}>mathbf{0}, boldsymbol{g}(boldsymbol{x})=|boldsymbol{x}| forall boldsymbol{x} leq mathbf{0} ) and
( h(x)=4 ) is-
( A cdot frac{7}{3} )
B. 9
c. 13
D. ( frac{40}{3} )
12
386 Find the area of bounded by ( y=sin x ) from ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}} ) to ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{2}} )
A ( cdot frac{sqrt{2}-1}{sqrt{2}} )
B. ( frac{1}{2} )
( c cdot frac{1}{4} )
D. None of these
12
387 The area of the region between the curve
( boldsymbol{y}=mathbf{4} boldsymbol{x}^{2} ) and the line ( boldsymbol{y}=mathbf{6} boldsymbol{x}-mathbf{2} ) is:
A ( cdot frac{1}{9} ) sq. units
B. ( frac{1}{12} ) sq. units
c. ( frac{3}{2} ) sq. units
D. ( frac{1}{5} ) sq. units
12
388 Find the area of the region in the first quadrant enclosed by x-axis, the line ( x=sqrt{3} y ) and the circle ( x^{2}+y^{2}=4 ) 12
389 Sketch for ( boldsymbol{y}=cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right) ) 12
390 Suppose ( g(x)=2 x+1 ) and ( h(x)= )
( 4 x^{2}+4 x+5 ) and ( h(x)=(f o g)(x) ) The
area enclosed by the graph of the function ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and the pair of
tangents drawn to it from the origin is
( A cdot 8 / 3 )
B. ( 16 / 3 )
c. ( 32 / 3 )
D. none
12
391 Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{3}, boldsymbol{y}=mathbf{3} boldsymbol{x}-mathbf{1} ) 12
392 The area of the region bounded by the parabola ( (boldsymbol{y}-mathbf{2})^{2}=(boldsymbol{x}-mathbf{1}), ) the
tangent to the parabola at the point (2,3) and the X-axis is?
( A cdot 3 )
B. 6
( c .9 )
D. 12
12
393 Find the area of the region bounded by
the curve ( y^{2}=4 x, y-a x i s & ) the line
( boldsymbol{y}=mathbf{3} )
12
394 Find the area of the region bounded by the curve ( y^{2}=4 x ) and the line ( x=3 )
begin{tabular}{l}
A. ( 4 sqrt{3} ) \
hline
end{tabular}
B. ( 8 sqrt{3} )
( c cdot 6 )
D. ( 2 sqrt{3} )
12
395 Find the area bounded by the curve ( y= )
( 2 x-x^{2}, ) and the line ( y=x )
12
396 If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi} ) and ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) are
angles of ( triangle ; ) then ( sin A+sin B+sin C )
is
( ^{A}=frac{3 sqrt{3}}{2} )
B. ( frac{3 sqrt{3}}{2} )
D. none of these
12
397 The area of the region, enclosed by the circle ( x^{2}+y^{2}=2 ) which is not common
to the region bounded by the parabola ( y^{2}=x ) and the straight line ( y=x, ) is:
A ( cdot frac{1}{3}(15 pi-1) )
в. ( frac{1}{6}(24 pi-1) )
c. ( frac{1}{6}(12 pi-1) )
D. ( frac{1}{3}(6 pi-1) )
12
398 The area of the region bounded by ( y= )
( -1, y=2, x=y^{3} ) and ( x=0 ) is
A ( cdot frac{17}{4} ) sq. units
B. ( frac{1}{4} ) sq. units
c. 4 sq. units
D. None of these
12
399 The area of the region bounded by the
curve ( boldsymbol{y}=boldsymbol{x}^{3}, ) its tangent at ( (mathbf{1}, mathbf{1}) ) and ( boldsymbol{x} )
-axis, is
A ( cdot frac{1}{12} ) sq unit
B ( cdot frac{1}{6} ) sq unit
c. ( frac{2}{17} ) sq unit
D. ( frac{2}{15} ) sq unit
12
400 Find the area bounded by the curves ( boldsymbol{y}=sqrt{1-boldsymbol{x}^{2}} ) and ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{x} . ) Also find
the ratio in which the y-axis divide this
area
A ( cdot frac{pi}{2} ; frac{pi-1}{pi+1} )
В. ( frac{pi}{4} ; frac{pi-1}{pi+1} )
c. ( frac{pi}{2} ; frac{pi+1}{pi-1} )
D. None of these
12
401 Find the area bounded by the parabola
( x^{2}=4 y ) and the straight line ( x=4 y- )
2
12
402 The area lying in the first quadrant between the curves ( x^{2}+y^{2}=pi^{2} ) and
( y=sin x ) and ( y- ) axis is
( ^{A} cdot frac{pi^{3}-8}{4} ) sq. units
B. ( frac{pi^{3}+8}{4} ) sq. units
c. ( 4left(pi^{3}-8right) ) sq. units
D. ( frac{pi-8}{4} ) sq. units
12
403 The area of the region bounded by the
( operatorname{lines} y=2 x+1, y=3 x+1 ) and ( x=4 )
is
A. 16 sq.unit
в. ( frac{121}{3} ) sq.unit
c. ( frac{121}{6} ) sq.unit
D. 8 sq.unit
12
404 37. The area enclosed by the curves y=sin x + cos x and
y= |cos X-sin x over the interval
(JEE Adv. 2013)
(a) 4(2-1) (b) 2.12 (12-1)
(C) 26/2 + 1) (d) 2/2(√2+1)
12
405 Find the area of the region bounded by
the curve ( y=x^{2} ) and the line ( y=2 )
12
406 The area bounded by curves 12
407 Area bounded by curve ( xleft(x^{2}+pright)= )
( boldsymbol{y}-mathbf{1} ) with ( boldsymbol{y}=mathbf{1} boldsymbol{p}<mathbf{0} ) is
A ( cdot frac{p^{2}}{4} )
в. ( frac{p}{2} )
c. ( frac{p^{2}}{2} )
D. ( frac{p}{4} )
12
408 Which of the following equation shows the above graph?
A ( cdot f(x)=x^{2}+9 )
B. ( f(x)=(x-9)^{2} )
( mathbf{c} cdot f(x)=9-x^{2} )
D. ( f(x)=-left|x^{2}+9right| )
E ( cdot f(x)=mid-x^{2}+9 )
12
409 The ratio of the area’s bounded by the curves ( y^{2}=12 x ) and ( x^{2}=12 y ) is
divided by the line ( x=3 ) is
A .15: 49
B. 9: 15
( c cdot 7: 15 )
D. 7:5
12
410 Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=mathbf{1}+boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2} ) 12
411 The area bounded by the two curves ( boldsymbol{y}=sin boldsymbol{x}, boldsymbol{y}=cos boldsymbol{x} ) and the X-axis in
the first quadrant ( left[0, frac{pi}{2}right] ) is
A ( .2-sqrt{2} ) sq. units
B . ( 2+sqrt{2} ) sq. units
c. ( 2(sqrt{2}-1) ) sq. units
D. 4 sq. units
12
412 The normals to the curve ( y=x^{2}-x+ )
1, drawn at the points with the abscissa
( boldsymbol{x}_{1}=mathbf{0}, boldsymbol{x}_{2}=-mathbf{1} ) and ( boldsymbol{x}_{3}=frac{mathbf{5}}{mathbf{2}} )
A. are parallel to each other
B. are pair wise perpendicular
c. are concurrent
D. are not concurrent
12
413 If ( A_{n} ) is the area bounded by ( y= )
( left(1-x^{2}right)^{n} ) and coordinate axes, ( n ) is in
set of natural numbers, then
A. ( A_{n}=A_{n-1} )
В. ( A_{n}A_{n-1} )
D. ( A_{n}=2 A_{n-1} )
12
414 Find the point on the curve ( 9 y^{2}=x^{3} )
where normal to the curve has non zero
x-intercept and both x-intercept and ( y ) intercept are equal
12
415 The figure shows a portions of the graph
( y=2 x-4 x^{3} . ) The line ( y=c ) is such that
the areas of the regions marked I and II are equal. If a,b are the ( x ) -coordinates of A,B respectively, then a + b equals
( A cdot frac{2}{sqrt{7}} )
B. ( frac{3}{sqrt{7}} )
( c cdot frac{4}{sqrt{7}} )
( D cdot frac{5}{sqrt{7}} )
12
416 If ( C_{1}={x: 1<x<2} ) and ( C_{2}={x: )
( 4<x<5}, ) find ( Pleft(C_{1} cup C_{2}right) )
12
417 Area bounded by curve ( y=x^{2} ) and ( y= )
( 2-x^{2} ) is ( ? )
A ( cdot frac{8}{3} ) sq units
B. ( frac{3}{8} ) sq units
c. ( frac{3}{2} ) sq units
D. None of these
12
418 Show that the height of the cylinder of maximum volume that can be inscribed
( boldsymbol{h} )
in a cone of height his
12
419 Estimating the absolute value of the ntegral ( int_{10}^{19} frac{sin x}{1+x^{8}} d x ) we get ( 10^{-m} ).Find
( boldsymbol{m} ? )
12
420 Find the area bounded by curves
( left{(x, y): y geq x^{2} text { and } y=|x|right} )
12
421 Semicircles are drawn outside by taking every side of regular hexagon as a diameter. The perimeter of hexagon is ( 60 mathrm{cm} . ) Find the area of complete figure formed as such. ( (pi=3.14)(sqrt{3}=1.73) )
A ( .495 mathrm{cm}^{2} )
B. 259.5cm ( ^{2} )
c. ( 235.5 mathrm{cm}^{2} )
D. ( 695.5 mathrm{cm}^{2} )
12
422 70. The area of the region between the curves y =
1+ sinx
COS X
1-sin x
and y= 1 A bounded by the lines x = 0 and x = 7
V cos x
(2008)
√2-1
√2-1
t.
Ő (1+t2)/1-42 ”
(b)
[
V2+1 27
– dt
(1+2 WI-42″
(d)
(d)
I
(©)
(+12) 1-2 ***
12
423 The area enclosed between the curves
( mathbf{y}=mathbf{a x}^{2} ) and ( mathbf{x}=mathbf{a y}^{2}(mathbf{a}>mathbf{0}) ) is 1 sq. unit,
then the value of a is
A. ( 1 / sqrt{3} )
в. ( 1 / 2 )
c. 1
D. ( 1 / 3 )
12
424 OLI 10 IUIDUOLLON
40.
The area (in square units) bounded by the curves
y= Vx, 2y-x+3=0, x-axis, and lying in the first quadrant
[JEEM 2013]
27
(a) 9 (6) 36 (c) 18 (d) –
is:
12
425 The area bounded by ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=[boldsymbol{x}+ )
1], ( x leq 1 ) and the ( y ) -axis is
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
( c cdot 1 )
D.
12
426 Parabolas ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{a}left(boldsymbol{x}-boldsymbol{c}_{1}right) ) and ( boldsymbol{x}^{2}= )
( 4 aleft(y-c_{2}right), ) where ( c_{1} ) and ( c_{2} ) are variable
are such that they touch each other. Locus of their point of contact is
A ( . x y=2 a^{2} )
В . ( x y=4 a^{2} )
c. ( x y=a^{2} )
D. none of these
12
427 Area lying between the curve ( y^{2}=4 x )
and ( boldsymbol{y}=mathbf{2} boldsymbol{x} ) is :
A ( cdot frac{2}{3} )
B. ( frac{1}{3} )
( c cdot frac{1}{4} )
D. ( frac{3}{4} )
12
428 Let ( A_{n} ) be the constant number such
that ( c>1 . ) If the least area of the figure
given by the line passing through the point (1,c) with gradient ‘m’ and the parabola ( boldsymbol{y}=boldsymbol{x}^{2} ) is 36 sq.units find the
value of ( left(c^{2}+m^{2}right) )
A ( cdot 104 )
B. 105
( c cdot 10 )
D. 100
12
429 Area of the circle ( (x-2)^{2}+(y-3)^{2}= )
32 which lies below the line ( y=x+1 ) is ( int_{-2}^{6}[(x+1)+sqrt{32-(x-2)^{2}}+3] d x )
12
430 The area of the region bounded by ( y= )
( boldsymbol{x}^{2}+mathbf{2}, boldsymbol{y}=-boldsymbol{x}, boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=mathbf{1} ) is
A ( cdot frac{17}{6} )
в. ( frac{17}{3} )
c. ( frac{10}{3} )
D. ( frac{8}{3} )
12
431 The area of the region bounded by the curves ( mathbf{y}=sqrt{boldsymbol{x}} ) and ( boldsymbol{y}=sqrt{mathbf{4}-mathbf{3} boldsymbol{x}} ) and
( mathbf{y}=mathbf{0} ) is:
A . ( 4 / 9 )
B. 16/9
( c cdot 8 / 9 )
D. 9/2
12
432 The area bounded by the ( x- ) axis, the
curve ( y=f(x) ) and the lines ( x=1 ) and
( x=b ) is equal to ( (sqrt{b^{2}+1}-sqrt{2}) ) for all
( boldsymbol{b}>1, ) then ( boldsymbol{f}(boldsymbol{x}) ) is
A. ( sqrt{x-1} )
B. ( sqrt{x+1} )
c. ( sqrt{x^{2}+1} )
D. ( frac{x}{sqrt{x^{2}+1}} )
12
433 Circular arc of radius ( 7 mathrm{cm} ) has been
drawn with vertex ( A ) of an equilateral
triangle ( A B C ) of side ( 14 mathrm{cm} ) at center Find the area of shaded region.
12
434 The value of ( a ) for which the area
between the curves ( y^{2}=4 a x ) and ( x^{2}= )
( 4 a y ) is ( 1 s q . ) unit, is-
A. ( sqrt{3} )
B. 4
c. ( 4 sqrt{3} )
D. ( frac{sqrt{3}}{4} )
12
435 The area of the region bounded by ( 3 x pm )
( 4 y pm 6=0 ) in sq.units is
( A cdot 3 )
в. 1.5
c. 4.5
D. 6
12
436 The area bounded by curve ( y=|x-1| )
and ( y=1 ) is
A .
B. 2
c. ( 1 / 2 )
D. None of these
12
437 Sketch the curve for ( sin y=sin x ) 12
438 Find the area of the smaller region bounded by the ellipse ( frac{x^{2}}{9}+frac{y^{2}}{4}=1 ) and the line ( frac{x}{3}+frac{y}{2}=1 ) 12
439 The area of the region bounded by the curve ( y=x^{2} ) and the line ( y=16 ) is
A ( cdot frac{128}{3} ) sq.units
в. ( frac{64}{3} ) sq.units
c. ( frac{32}{3} ) sq.units
D. ( frac{256}{3} ) sq.units
12
440 Area of ( triangle A B C=68.2 s q m . ) Find the
area of shaded region
12
441 The point intersection of the tangents drawn to the curve ( x^{2} y=1-y ) at the
points where it is met by the curve ( x y=1-y ) is given by
A ( .(0,1) )
в. (1,1)
c. (1,0)
(i) 5
D. ( (0, infty) )
12
442 Area of the region bounded by the curve
( boldsymbol{y}=mathbf{2 5}^{boldsymbol{x}}+mathbf{1 6} ) and curve ( boldsymbol{y}=boldsymbol{b} . mathbf{5}^{boldsymbol{x}}+mathbf{4} )
whose tangent at the point ( boldsymbol{x}=mathbf{1} )
makes an angle ( tan ^{-1}(40 log 5) ) with
the ( x- ) axis is:
( ^{mathbf{A}} cdot_{2 log _{5}}left(frac{e^{4}}{27}right) )
( ^{mathbf{B}} cdot_{4 log _{5}}left(frac{e^{4}}{27}right) )
( mathbf{c} cdot_{3 log _{5}}left(frac{e^{4}}{27}right) )
D. None of these
12
443 3. Sketch the region whose area is represented by the definite
integral and evaluate the integral using an appropriate
formula form geometry.
(a) f2dx
(b) } (x + 2) dx
(c) | (x – 1) dx
12
444 The area (in sq. units) of the region described by
( left{(x, y) ; y^{2} leq 2 x text { and } y geq 4 x-1right} ) is
( A cdot frac{7}{32} )
в. ( frac{5}{64} )
c. ( frac{15}{64} )
D. ( frac{9}{32} )
12
445 37.
The area between the parabolas
straight line y=2 is:
las r? = and r2 =9y and the
[2012]
() 1012
@ 2012 (6) 102 () 2012
12
446 Sketch the graph for ( boldsymbol{y}= ) ( log _{frac{1}{4}}left(x-frac{1}{4}right)+ )
( frac{1}{2} log _{4}left(16 x^{2}-8 x+1right) )
12
447 The curve whose subtangent is twice the abscissa of the point of contact and passing through (1,2) is
A ( cdot y^{2}=4 x )
B ( cdot y^{2}=-4 x )
c. ( x^{2}=4 y )
D. ( x^{2}=-4 y )
12
448 Find the area bounded by curves
( (x-1)^{2}+y^{2}=1 ) and ( x^{2}+y^{2}=1 )
12
449 Find the axis of symmetry of the
parabola shown
A ( . x=1 )
в. ( x=2.5 )
( mathbf{c} cdot x=3 )
D. None of these
12
450 Let ( boldsymbol{f}(boldsymbol{x})=-boldsymbol{x}^{2} / 2 . ) If the graph of ( boldsymbol{f}(boldsymbol{x}) )
is translated 2 units up and 3 units left and the resulting graph is that of ( g(x) ) ( operatorname{then} g(1 / 2)= )
A .
B . ( -1 / 8 )
c. ( -2 / 8 )
D. ( -33 / 8 )
E . ( 13 / 8 )
12
451 41. Area of the region {(x,y) eR?:y>x +3], 5y 5x+9515}
is equal to
(JEE Adv. 2016)
12
452 If ( theta leq x leq pi ; ) then the area bounded by
the curve ( boldsymbol{y}=boldsymbol{x} ) and ( boldsymbol{y}=boldsymbol{x}+sin boldsymbol{x} ) is
A . 2
B. 4
( c cdot 2 pi )
D. ( 4 pi )
12
453 If the area (in sq. units) bounded by the parabola ( y^{2}=4 lambda x ) and the line ( y= ) ( lambda x, lambda>0, ) is ( frac{1}{9}, ) then ( lambda ) is equal to
A .24
B . 48
( c cdot 4 sqrt{3} )
D. ( 2 sqrt{6} )
12
454 é tet
5.
For any real t, x=9
et – et
– is a point on the
hyperbola x2 – y2 = 1. Show that the area bounded by this
hyperbola and the lines joining its centre to the points
corresponding to t, and -t, is t,
(1982 – 3 Marks)
12
455 The area (in square units) of the region
bounded by ( boldsymbol{x}=-mathbf{1}, boldsymbol{x}=mathbf{2}, boldsymbol{y}=boldsymbol{x}^{2}+mathbf{1} )
and ( y=2 x-2 ) is
A . 10
B. 7
( c cdot 8 )
D.
12
456 40. Let b = 0 and for j = 0, 1, 2, …, n, let S, be the area of the
region bounded by the y-axis and the curve xeay = sin by,
jrcc (+1). Show that S, S,, S., …, S, are in
ь – у – ь
geometric progression. Also, find their sum for a=-1 and
b=n.
(2001 – 5 Marks)
12
457 The positive value of parameter’ ( a ) ‘ for
which the area bounded by parabolas ( y=x-a x^{2} & a y=x^{2} ) attains its the
maximum value is
12
458 Area lying in the first quadrant and bounded by the circle ( x^{2}+y^{2}=4 ) and
the line ( boldsymbol{x}=boldsymbol{y} sqrt{mathbf{3}} ) is:
A . ( pi )
в. ( frac{pi}{2} )
c. ( frac{pi}{3} )
D. None of these
12
459 4. Find the area under the curve y = cos x over
(a) [0, 7/2] (b) [0, 1]
curve y = cos x over the interval
AY
y = cos x
-1+
12
460 Find the area of the region
( left{(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4}, boldsymbol{x}+boldsymbol{y} geq mathbf{2}right} )
12
461 Find the area of the region enclosed by the curves;
( boldsymbol{y}=cos boldsymbol{x}, boldsymbol{y}=1-frac{2 x}{pi} )
A ( cdot 2-frac{pi}{2} )
в. ( pi )
c. ( frac{pi}{2} )
D.
12
462 Sketch the graph for ( y=cos ^{-1}(cos x) ) 12
463 The area bounded by the curves ( y= )
( cos x ) and ( y=sin x ) between the ordinates ( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=frac{mathbf{3} boldsymbol{pi}}{mathbf{2}} ) is
A. ( (4 sqrt{2}-2) ) sq units
B . ( (4 sqrt{2}+2) ) sq units
c. ( (4 sqrt{2}-1) ) sq units
D. ( (4 sqrt{2}+1) ) sq units
12
464 Find the area of the region bounded by
the curve ( y^{2}=x ) and the lines ( x= )
( 1, x=4 ) and the ( x ) -axis
( A cdot frac{8}{3} )
в. ( frac{14}{3} )
( c cdot frac{7}{3} )
D. ( frac{1}{3} )
12
465 The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= )
( mathbf{1} )
A ( cdot frac{4}{3} )
B.
( c cdot frac{2}{3} )
D.
12
466 Let ( P(x, y) ) be a moving point in the ( x- ) ( boldsymbol{y} ) plane such that ( [boldsymbol{x}] .[boldsymbol{y}]=2, ) where ( [.] )
denotes the greatest integer function, then area of the region containing the points ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) is equal to:
A. 1 sq. units
B. 2 sq. units
c. 4 sq. units
D. None of these
12
467 If a curve passes through the point ( left(2, frac{7}{2}right) ) and has slope ( left(1-frac{1}{x^{2}}right) ) at any
point ( (x, y) ) on it, then the abscissa of the point on the curve whose ordinate is
( frac{-3}{2} ) is
( A cdot 2 )
B. -2
c. 1
D. –
12
468 The area enclosed between the curve
( boldsymbol{y}^{2}=boldsymbol{x} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{y}=|boldsymbol{x}| ) is
A ( cdot frac{1}{6} )
в. ( frac{1}{3} )
( c cdot frac{2}{3} )
D. 1
12
469 Find the area bounded by the curves ( boldsymbol{x}=boldsymbol{a} cos boldsymbol{t}, boldsymbol{y}=boldsymbol{b} sin t ) in the first
quadrant
A ( cdot frac{pi a b}{4} )
B. ( frac{pi a^{2} b}{4} )
c. None of these
D. ( frac{pi a b^{2}}{4} )
12
470 Area bounded by the lines ( y=x, x= )
( -1, x=2 ) and ( x ) -axis is
A ( cdot frac{5}{2} ) sq. units
B. ( frac{3}{2} ) squnits
c. ( frac{1}{2} s q ) units
D. None of these
12
471 ( * )
( vdots )
( * )
( * )
12
472 The ratio in which the area bounded by the curves ( y^{2}=12 x ) and ( x^{2}=12 y ) is
divided by the line ( x=3 ) is
A . 15: 16
B. 15: 49
( c cdot 1: 2 )
D. None of these
12
473 The area bounded by the parabolas ( y^{2}=4 a(x+a) ) and ( y^{2}=-4 a(x-a) )
is
A ( cdot frac{16}{3} a^{2} ) sq units
B. ( frac{8}{3} ) sq units
c. ( frac{4}{3} a^{2} ) sq units
D. None of these
12
474 Compute the area of the figure which lies in the first quadrant inside the
curve ( x^{2}+y^{2}=3 a^{2} & ) is bounded by
the parabola ( x^{2}=2 a y & y^{2}= )
( 2 a x(a>0) )
12
475 If the curves ( y=x^{3}+a x ) and ( y= )
( b x^{2}+c ) pass through the point (-1,0)
and have common tangent line at this
point, then the value of ( a+b ) is?
( mathbf{A} cdot mathbf{0} )
в. -2
c. -3
D. –
12
476 What is the area of a plane figure bounded by the points of the lines max ( (x, y)=1 ) and ( x^{2}+y^{2}=1 ? ) 12

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