Area Under The Curves
161 Questions
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Evening Shift
The area of the region bounded by y $-$ x = 2 and x2 = y is equal to :
A.
${{16} \over 3}$
B.
${{2} \over 3}$
C.
${{9} \over 2}$
D.
${{4} \over 3}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 27th July Morning Shift
If the area of the bounded region
$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$ is ,
$\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $, then the value of ${(\alpha + \beta - 2\lambda )^2}$ is equal to :
$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$ is ,
$\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $, then the value of ${(\alpha + \beta - 2\lambda )^2}$ is equal to :
A.
8
B.
2
C.
4
D.
1
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Morning Shift
The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $ is :
A.
${8 \over 3}$
B.
${{17} \over 3}$
C.
${{13} \over 3}$
D.
${7 \over 3}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Evening Shift
The area bounded by the curve 4y2 = x2(4 $-$ x)(x $-$ 2) is equal to :
A.
${\pi \over {16}}$
B.
${\pi \over {8}}$
C.
${3\pi \over {2}}$
D.
${3\pi \over {8}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = ${\pi \over 2}$ in the first quadrant. Then,
A.
${A_1}:{A_2} = 1:\sqrt 2 $ and ${A_1} + {A_2} = 1$
B.
${A_1} = {A_2}$ and ${A_1} + {A_2} = \sqrt 2 $
C.
$2{A_1} = {A_2}$ and ${A_1} + {A_2} = 1 + \sqrt 2 $
D.
${A_1}:{A_2} = 1:2$ and ${A_1} + {A_2} = 1$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Evening Shift
The area of the region : $R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $ is :
A.
$6\sqrt 3 $ square units
B.
$12\sqrt 3 $ square units
C.
$11\sqrt 3 $ square units
D.
$9\sqrt 3 $ square units
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Morning Shift
The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola
y2 = 9x, is :
A.
$12\pi - 3\sqrt 3 $
B.
$24\pi + 3\sqrt 3 $
C.
$24\pi - 3\sqrt 3 $
D.
$12\pi + 3\sqrt 3 $
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 31st August Evening Shift
If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = ${3 \over 2}$ and the curve y = 1 + 4x $-$ x2, then 12 m is equal to _____________.
Correct Answer: 26
Explanation:
According to the question,
${1 \over 2}\int_0^{3/2} {(1 + 4x - {x^2})dx = \int_0^{3/2} {mx\,dx} } $
$ \Rightarrow {1 \over 2}\left[ {\left( {x + 2{x^2} - {{{x^3}} \over 3}} \right)} \right]_0^{3/2} = {m \over 2}[x]_0^{3/2} \Rightarrow {3 \over 2} + {9 \over 2} - {9 \over 8} = {{9m} \over 4}$
$\Rightarrow$ m = 39/18 $\Rightarrow$ 12m = 26
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th August Evening Shift
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 $-$ 3x2 $-$ 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______________.
Correct Answer: 114
Explanation:
f'(x) = 6x2 $-$ 6x $-$ 12 = 6(x $-$ 2) (x + 1)
Point = (2, $-$20) & ($-$1, 7)
$A = \int\limits_{ - 1}^0 {(2{x^3} - 3{x^2} - 12x)dx + \int\limits_0^2 {(12x + 3{x^2} - 2{x^3})\,dx} } $
$A = \left( {{{{x^4}} \over 2} - {x^3} - 6{x^2}} \right)_{ - 1}^0 + \left( {6{x^2} + {x^3} - {{{x^4}} \over 2}} \right)_0^2$
4A = 114
Point = (2, $-$20) & ($-$1, 7)
$A = \int\limits_{ - 1}^0 {(2{x^3} - 3{x^2} - 12x)dx + \int\limits_0^2 {(12x + 3{x^2} - 2{x^3})\,dx} } $
$A = \left( {{{{x^4}} \over 2} - {x^3} - 6{x^2}} \right)_{ - 1}^0 + \left( {6{x^2} + {x^3} - {{{x^4}} \over 2}} \right)_0^2$
4A = 114
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th August Morning Shift
The area of the region $S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\} $ is ____________.
Correct Answer: 27
Explanation:
For A & B
3x2 = 6x + 24 $\Rightarrow$ x2 $-$ 2x $-$ 8 = 0
$\Rightarrow$ x = $-$2, 4
Area $ = \int\limits_{ - 2}^4 {\left( {{3 \over 2}x + 6 - {3 \over 4}{x^2}} \right)dx} $
$ = \left[ {{{3{x^2}} \over 4} + 6x - {{{x^3}} \over 4}} \right]_{ - 2}^4 = 27$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 22th July Evening Shift
The area (in sq. units) of the region bounded by the curves x2 + 2y $-$ 1 = 0, y2 + 4x $-$ 4 = 0 and y2 $-$ 4x $-$ 4 = 0, in the upper half plane is _______________.
Correct Answer: 2
Explanation:
Required area (shaded)
$ = 2\left[ {\int\limits_0^2 {\left( {{{4 - {y^2}} \over 4}} \right)dy - \int\limits_0^1 {\left( {{{1 - {x^2}} \over 2}} \right)dx} } } \right]$
$ = 2\left[ {{4 \over 3} - {1 \over 3}} \right] = (2)$
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 20th July Morning Shift
Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 and x = $\sqrt 5 $ is $\alpha$$\sqrt 5 $ + $\beta$ + $\gamma$ cos$-$1$\left( {{1 \over {\sqrt 5 }}} \right)$, then |$\alpha$ + $\beta$ + $\gamma$| is equal to ______________.
Correct Answer: 1.25
Explanation:
E : x2 + 4y2 = 5
Tangent at P : x + 4y = 5
Required area
$ = \int\limits_1^{\sqrt 5 } {\left( {{{5 - x} \over 4} - {{\sqrt {5 - {x^2}} } \over 2}} \right)dx} $
$ = \left[ {{{5x} \over 4} - {{{x^2}} \over 8} - {x \over 4}\sqrt {5 - {x^2}} - {5 \over 2}{{\sin }^{ - 1}}{x \over {\sqrt 5 }}} \right]_1^{\sqrt 5 }$
$ = {5 \over 4}\sqrt 5 - {5 \over 4} - {5 \over 4}{\cos ^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)$
If we assume $\alpha$, $\beta$, $\gamma$, $\in$ Q (Not given in question) then $\alpha$ = ${5 \over 4}$, $\beta$ = $-$${5 \over 4}$ & $\gamma$ = $-$${5 \over 4}$
|$\alpha$ + $\beta$ + $\gamma$| = 1.25
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 17th March Evening Shift
Let f : [$-$3, 1] $ \to $ R be given as
$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.$
If the area bounded by y = f(x) and x-axis is A, then the value of 6A is equal to ___________.
$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x^2}\} ,\,0 \le x \le 1. \hfill \cr} \right.$
If the area bounded by y = f(x) and x-axis is A, then the value of 6A is equal to ___________.
Correct Answer: 41
Explanation:
Area is $\int\limits_{ - 3}^{ - 2} {(x + 6)dx + \int\limits_{ - 2}^0 {{x^2}dx + \int\limits_0^1 {\sqrt {x}dx = A} } } $
$ = {7 \over 2} + \left[ {{{{x^3}} \over 3}} \right]_{ - 2}^0 + \left[ {{2 \over 3}{x^{3/2}}} \right]_0^1$
$ = {7 \over 2} + {8 \over 3} + {2 \over 3} = {{41} \over 6}$
So, 6A = 41
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 26th February Morning Shift
The area bounded by the lines y = || x $-$ 1 | $-$ 2 | is ___________.
Correct Answer: 8
Explanation:
Question is incomplete it should be area bounded
by y = || x $-$ 1 | $-$ 2 | and y = 2.
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Morning Shift
The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to __________.
Correct Answer: 64
Explanation:
$A = \int\limits_{{\pi \over 4}}^{{{5\pi } \over 4}} {(\sin x - \cos x)dx} $
$= [ - \cos x - \sin x]_{\pi /4}^{5\pi /4}$
$ = - \left[ {\left( {\cos {{5\pi } \over 4} + \sin {5\pi \over 4}} \right) - \left( {\cos {\pi \over 4} + \sin {\pi \over 4}} \right)} \right]$
$ = - \left[ {\left( { - {1 \over {\sqrt 2 }} - {1 \over {\sqrt 2 }}} \right) - \left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}} \right)} \right]$
$ = {4 \over {\sqrt 2 }} = 2\sqrt 2 $
$ \Rightarrow {A^4} = {\left( {2\sqrt 2 } \right)^4} = 64$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
The area (in sq. units) of the region enclosed
by the curves y = x2 – 1 and y = 1 – x2 is equal to :
by the curves y = x2 – 1 and y = 1 – x2 is equal to :
A.
${8 \over 3}$
B.
${4 \over 3}$
C.
${7 \over 2}$
D.
${{16} \over 3}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Morning Slot
The area (in sq. units) of the region
A = {(x, y) : |x| + |y| $ \le $ 1, 2y2 $ \ge $ |x|}
A = {(x, y) : |x| + |y| $ \le $ 1, 2y2 $ \ge $ |x|}
A.
${1 \over 6}$
B.
${5 \over 6}$
C.
${1 \over 3}$
D.
${7 \over 6}$
For point of intersection
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Evening Slot
The area (in sq. units) of the region
A = {(x, y) : (x – 1)[x] $ \le $ y $ \le $ 2$\sqrt x $, 0 $ \le $ x $ \le $ 2}, where [t]
denotes the greatest integer function, is :
A = {(x, y) : (x – 1)[x] $ \le $ y $ \le $ 2$\sqrt x $, 0 $ \le $ x $ \le $ 2}, where [t]
denotes the greatest integer function, is :
A.
${8 \over 3}\sqrt 2 - 1$
B.
${4 \over 3}\sqrt 2 + 1$
C.
${8 \over 3}\sqrt 2 - {1 \over 2}$
D.
${4 \over 3}\sqrt 2 - {1 \over 2}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
The area (in sq. units) of the region
{ (x, y) : 0 $ \le $ y $ \le $ x2 + 1, 0 $ \le $ y $ \le $ x + 1,
${1 \over 2}$ $ \le $ x $ \le $ 2 } is :
{ (x, y) : 0 $ \le $ y $ \le $ x2 + 1, 0 $ \le $ y $ \le $ x + 1,
${1 \over 2}$ $ \le $ x $ \le $ 2 } is :
A.
${{79} \over {16}}$
B.
${{79} \over {24}}$
C.
${{23} \over {6}}$
D.
${{23} \over {16}}$
$A = \int\limits_{{1 \over 2}}^1 {({x^2} + 1)dx + } $$\int\limits_1^2 {} $${(x + 1)dx}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
Consider a region R = {(x, y) $ \in $ R : x2 $ \le $ y $ \le $ 2x}.
if a line y = $\alpha $ divides the area of region R into
two equal parts, then which of the following is
true?
A.
3$\alpha $2 - 8$\alpha $ + 8 = 0
B.
$\alpha $3 - 6$\alpha $3/2 - 16 = 0
C.
3$\alpha $2 - 8$\alpha $3/2 + 8 = 0
D.
$\alpha $3 - 6$\alpha $2 + 16 = 0
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
Area (in sq. units) of the region outside
${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$ and inside the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$ is :
${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$ and inside the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$ is :
A.
$6\left( {4 - \pi } \right)$
B.
$3\left( {4 - \pi } \right)$
C.
$6\left( {\pi - 2} \right)$
D.
$3\left( {\pi - 2} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Evening Slot
Given : $f(x) = \left\{ {\matrix{
{x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr
{{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr
{1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr
} } \right.$
and $g(x) = \left( {x - {1 \over 2}} \right)^2,x \in R$
Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x) and y = g(x) between the lines, 2x = 1 and 2x = $\sqrt 3 $, is :
and $g(x) = \left( {x - {1 \over 2}} \right)^2,x \in R$
Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x) and y = g(x) between the lines, 2x = 1 and 2x = $\sqrt 3 $, is :
A.
${1 \over 2} + {{\sqrt 3 } \over 4}$
B.
${1 \over 2} - {{\sqrt 3 } \over 4}$
C.
${1 \over 3} + {{\sqrt 3 } \over 4}$
D.
${{\sqrt 3 } \over 4} - {1 \over 3}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Evening Slot
The area (in sq. units) of the region
{(x,y) $ \in $ R2 : x2 $ \le $ y $ \le $ 3 – 2x}, is :
{(x,y) $ \in $ R2 : x2 $ \le $ y $ \le $ 3 – 2x}, is :
A.
${{34} \over 3}$
B.
${{29} \over 3}$
C.
${{31} \over 3}$
D.
${{32} \over 3}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Morning Slot
For a > 0, let the curves C1 : y2 = ax and
C2 : x2 = ay intersect at origin O and a point P.
Let the line x = b (0 < b < a) intersect the chord
OP and the x-axis at points Q and R,
respectively. If the line x = b bisects the area
bounded by the curves, C1 and C2, and the area of
$\Delta $OQR = ${1 \over 2}$, then 'a' satisfies the equation :
$\Delta $OQR = ${1 \over 2}$, then 'a' satisfies the equation :
A.
x6 – 12x3 + 4 = 0
B.
x6 – 12x3 – 4 = 0
C.
x6 + 6x3 – 4 = 0
D.
x6 – 6x3 + 4 = 0
C1 : y2
= ax, C2 : x2
= ay (a > 0)
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 7th January Evening Slot
The area (in sq. units) of the region
{(x, y) $ \in $ R2 | 4x2 $ \le $ y $ \le $ 8x + 12} is :
{(x, y) $ \in $ R2 | 4x2 $ \le $ y $ \le $ 8x + 12} is :
A.
${{125} \over 3}$
B.
${{128} \over 3}$
C.
${{127} \over 3}$
D.
${{124} \over 3}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 7th January Morning Slot
The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x, is:
A.
${1 \over 6}\left( {24\pi - 1} \right)$
B.
${1 \over 3}\left( {12\pi - 1} \right)$
C.
${1 \over 3}\left( {6\pi - 1} \right)$
D.
${1 \over 6}\left( {12\pi - 1} \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th April Evening Slot
If the area (in sq. units) bounded by the parabola y2
= 4$\lambda $x and the line y = $\lambda $x, $\lambda $ > 0, is ${1 \over 9}$
, then $\lambda $ is equal to :
A.
$4\sqrt 3 $
B.
2$\sqrt 6 $
C.
48
D.
24
y2 = 4$\lambda $x and y = $\lambda x$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th April Morning Slot
If the area (in sq. units) of the region {(x, y) : y2
$ \le $ 4x, x + y $ \le $ 1, x $ \ge $ 0, y $ \ge $ 0} is a $\sqrt 2 $ + b, then a – b is equal
to :
A.
${8 \over 3}$
B.
$ - {2 \over 3}$
C.
6
D.
${{10} \over 3}$
Let P be the point common to x + y = 1 and y2 = 4x
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Evening Slot
The area (in sq.units) of the region bounded by the curves y = 2x
and y = |x + 1|, in the first quadrant is :
A.
${1 \over 2}$
B.
${3 \over 2}$
C.
${3 \over 2} - {1 \over {\log _e^2}}$
D.
$\log _e^2 + {3 \over 2}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Evening Slot
The area (in sq. units) of the region
A = {(x, y) : ${{y{}^2} \over 2}$ $ \le $ x $ \le $ y + 4} is :-
A = {(x, y) : ${{y{}^2} \over 2}$ $ \le $ x $ \le $ y + 4} is :-
A.
30
B.
18
C.
${{53} \over 3}$
D.
16
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Morning Slot
The area (in sq. units) of the region
A = {(x, y) : x2 $ \le $ y $ \le $ x + 2} is
A = {(x, y) : x2 $ \le $ y $ \le $ x + 2} is
A.
${{31 \over 6}}$
B.
${{10 \over 3}}$
C.
${{13 \over 6}}$
D.
${{9 \over 2}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
Let S($\alpha $) = {(x, y) : y2
$ \le $ x, 0 $ \le $ x $ \le $ $\alpha $} and A($\alpha $)
is area of the region S($\alpha $). If for a $\lambda $, 0 < $\lambda $ < 4,
A($\lambda $) : A(4) = 2 : 5, then $\lambda $ equals
A.
$2{\left( {{4 \over {25}}} \right)^{{1 \over 3}}}$
B.
$2{\left( {{2 \over {5}}} \right)^{{1 \over 3}}}$
C.
$4{\left( {{4 \over {25}}} \right)^{{1 \over 3}}}$
D.
$4{\left( {{2 \over {5}}} \right)^{{1 \over 3}}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Morning Slot
The area (in sq. units) of the region
A = { (x, y) $ \in $ R × R| 0 $ \le $ x $ \le $ 3, 0 $ \le $ y $ \le $ 4, y $ \le $ x2 + 3x} is :
A = { (x, y) $ \in $ R × R| 0 $ \le $ x $ \le $ 3, 0 $ \le $ y $ \le $ 4, y $ \le $ x2 + 3x} is :
A.
${{59} \over 6}$
B.
${{26} \over 3}$
C.
8
D.
${{53} \over 6}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Morning Slot
The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is
A.
${{15} \over 4}$
B.
${{15} \over 2}$
C.
${{21} \over 2}$
D.
${{17} \over 4}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is :
A.
${8 \over 3}$
B.
${{14} \over 3}$
C.
${{187} \over {24}}$
D.
${{37} \over {24}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is :
A.
${3 \over 4}$
B.
${5 \over 4}$
C.
${7 \over 8}$
D.
${9 \over 8}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Morning Slot
If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is -
A.
$\sqrt 3 $
B.
${{\sqrt 3 } \over 2}$
C.
${2 \over {\sqrt 3 }}$
D.
${1 \over {\sqrt 3 }}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Evening Slot
The area of the region
A = {(x, y) : 0 $ \le $ y $ \le $x |x| + 1 and $-$1 $ \le $ x $ \le $1} in sq. units, is :
A = {(x, y) : 0 $ \le $ y $ \le $x |x| + 1 and $-$1 $ \le $ x $ \le $1} in sq. units, is :
A.
${2 \over 3}$
B.
2
C.
${4 \over 3}$
D.
${1 \over 3}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Morning Slot
The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is :
A.
$56\over3$
B.
$32\over3$
C.
$8\over3$
D.
$14\over3$
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 16th April Morning Slot
If the area of the region bounded by the curves, $y = {x^2},y = {1 \over x}$ and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal to :
A.
${e^{{3 \over 2}}}$
B.
${4 \over 3}$
C.
${3 \over 2}$
D.
${e^{{2 \over 3}}}$
2018
JEE Mains
MCQ
JEE Main 2018 (Offline)
Let g(x) = cosx2, f(x) = $\sqrt x $ and $\alpha ,\beta \left( {\alpha < \beta } \right)$ be the roots of the quadratic equation 18x2 - 9$\pi $x + ${\pi ^2}$ = 0. Then the area (in sq. units) bounded by the curve
y = (gof)(x) and the lines $x = \alpha $, $x = \beta $ and y = 0 is :
y = (gof)(x) and the lines $x = \alpha $, $x = \beta $ and y = 0 is :
A.
${1 \over 2}\left( {\sqrt 2 - 1} \right)$
B.
${1 \over 2}\left( {\sqrt 3 - 1} \right)$
C.
${1 \over 2}\left( {\sqrt 3 + 1} \right)$
D.
${1 \over 2}\left( {\sqrt 3 - \sqrt 2 } \right)$
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Morning Slot
The area (in sq. units) of the region
{x $ \in $ R : x $ \ge $ 0, y $ \ge $ 0, y $ \ge $ x $-$ 2 and y $ \le $ $\sqrt x $}, is :
{x $ \in $ R : x $ \ge $ 0, y $ \ge $ 0, y $ \ge $ x $-$ 2 and y $ \le $ $\sqrt x $}, is :
A.
${{13} \over 3}$
B.
${{8} \over 3}$
C.
${{10} \over 3}$
D.
${{5} \over 3}$
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is :
A.
${1 \over {2\sqrt 3 }} + {\pi \over 3}$
B.
${1 \over {\sqrt 3 }} + {{2\pi } \over 3}$
C.
${1 \over {2\sqrt 3 }} + {{2\pi } \over 3}$
D.
${1 \over {\sqrt 3 }} + {{4\pi } \over 3}$
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
The area (in sq. units) of the region
$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$ is
$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$ is
A.
${3 \over 2}$
B.
${7 \over 3}$
C.
${5 \over 2}$
D.
${59 \over 12}$
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 9th April Morning Slot
The area (in sq. units) of the region described by
A= {(x, y) $\left| {} \right.$y$ \ge $ x2 $-$ 5x + 4, x + y $ \ge $ 1, y $ \le $ 0} is :
A= {(x, y) $\left| {} \right.$y$ \ge $ x2 $-$ 5x + 4, x + y $ \ge $ 1, y $ \le $ 0} is :
A.
${7 \over 2}$
B.
${{19} \over 6}$
C.
${{13} \over 6}$
D.
${{17} \over 6}$
2016
JEE Mains
MCQ
JEE Main 2016 (Offline)
The area (in sq. units) of the region $\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$ is :
A.
$\pi - {{4\sqrt 2 } \over 3}$
B.
${\pi \over 2} - {{2\sqrt 2 } \over 3}$
C.
$\pi - {4 \over 3}$
D.
$\pi - {8 \over 3}$
2015
JEE Mains
MCQ
JEE Main 2015 (Offline)
The area (in sq. units) of the region described by
$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$ and $\left. {y \ge 4x - 1} \right\}$ is :
$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$ and $\left. {y \ge 4x - 1} \right\}$ is :
A.
${{15} \over {64}}$
B.
${{9} \over {32}}$
C.
${{7} \over {32}}$
D.
${{5} \over {64}}$
2014
JEE Mains
MCQ
JEE Main 2014 (Offline)
The area of the region described by
$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$ and $\left. {{y^2} \le 1 - x} \right\}$ is :
$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$ and $\left. {{y^2} \le 1 - x} \right\}$ is :
A.
${\pi \over 2} - {2 \over 3}$
B.
${\pi \over 2} + {2 \over 3}$
C.
${\pi \over 2} + {4 \over 3}$
D.
${\pi \over 2} - {4 \over 3}$
2013
JEE Mains
MCQ
JEE Main 2013 (Offline)
The area (in square units) bounded by the curves $y = \sqrt {x,} $ $2y - x + 3 = 0,$ $x$-axis, and lying in the first quadrant is :
A.
$9$
B.
$36$
C.
$18$
D.
${{27} \over 4}$
2012
JEE Mains
MCQ
AIEEE 2012
The area between the parabolas ${x^2} = {y \over 4}$ and ${x^2} = 9y$ and the straight line $y=2$ is :
A.
$20\sqrt 2 $
B.
${{10\sqrt 2 } \over 3}$
C.
${{20\sqrt 2 } \over 3}$
D.
$10\sqrt 2 $
