How many questions from Area Under Curves appear in JEE?

Area Under Curves has appeared in 139 of the last 186 JEE Main and JEE Advanced papers — about 75% of them — contributing 149 questions in total across those papers.

Is Area Under Curves an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 75% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Area Under Curves questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Area Under Curves JEE Previous Year Questions with Answers

Q1·MathematicsNumericalJEE Advanced 2026

Passage: Consider the ellipses given by

x^{2} + 4y^{2} = 1

and

4x^{2} + y^{2} = 1

. Question: If

\alpha

is the area of the common region that lies inside both the given ellipses, then the value of

\cot\alpha

is _____.

Q2·MathematicsNumericalJEE Advanced 2026

Passage: Consider the curve

C_{1}

given by

y = e^{-x}

for

x \in [0, 10\pi]

, and the curve

C_{2}

given by

y = e^{-x}(\sin x + \cos x)

for

x \in [0, 10\pi]

. Let

n

be the total number of points of intersection of the curves

C_{1}

and

C_{2}

. Suppose that

\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} \in [0, 10\pi]

are the

x

-coordinates of the points of intersection of the curves

C_{1}

and

C_{2}

such that

\alpha_{1} < \alpha_{2} < \cdots < \alpha_{n}

. Question: Let

\beta

be the area of the region enclosed between the curves

C_{1}

,

C_{2}

, and the lines

x = \alpha_{1}

and

x = \alpha_{4}

. Then the value of

-\dfrac{1}{\pi}\log_{e}\left(\beta - 2e^{-\frac{\pi}{2}}\right)

is _____.

Q3·MathematicsSingle correctJEE Main 2026

Let

f : (1, \infty) \to \mathbb{R}

be a function defined as

f(x) = \dfrac{x-1}{x+1}

. Let

f^{i+1}(x) = f(f^{i}(x))

,

i = 1, 2, \ldots, 25

, where

f^{1}(x) = f(x)

. If

g(x) + f^{26}(x) = 0

,

x \in (1, \infty)

, then the area of the region bounded by the curves

y = g(x)

,

2y = 2x - 3

,

y = 0

and

x = 4

is:

(A) $\dfrac{1}{8} + \log_e 2$
(B) $\dfrac{1}{4} + \log_e 2$
(C) $\dfrac{5}{6} + 3\log_e 2$
(D) $\dfrac{5}{6} + \log_e 2$

Q4·MathematicsSingle correctJEE Main 2026

The area of the region

\{(x, y) : x^{2} - 8x \le y \le -x\}

is :

(A) $\frac{343}{6}$
(B) $\frac{637}{6}$
(C) $\frac{437}{6}$
(D) $\frac{523}{6}$

Q5·MathematicsSingle correctJEE Main 2026

The area of the region {(x, y) : 0 ≤ y ≤ 6 − x, y

^2

≥ 4x − 3, x ≥ 0} is:

(A)8
(B)9
(C)12
(D)15

Q6·MathematicsSingle correctJEE Main 2026

Let e be the base of natural logarithm and let f : {1, 2, 3, 4} → {1, e, e

^2

, e

^3

} and g : {1, e, e

^2

, e

^3

} →

\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right\}

be two bijective functions such that f is strictly decreasing and g is strictly increasing. If

\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\frac{1}{2}\right)\right\}\right]^{x}

, then the area of the region R = {(x, y) : x

^2

≤ y ≤ φ(x), 0 ≤ x ≤ 1} is:

(A) $\frac{3 - \log_e(2)}{3\log_e(2)}$
(B) $\frac{1}{3\log_e(2)}$
(C)3 + log $_e$ (2)
(D) $\frac{3 + \log_e(2)}{2 + \log_e(3)}$

Q7·MathematicsNumericalJEE Main 2026

Let f : ℝ → ℝ be a function such that

f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x

, x ∈ ℝ. Let the maximum value of f on ℝ be α. If the area of the region bounded by the curves

g(x) = x^{2}

and

h(x) = \beta x^{3}

, β > 0, is

\alpha^{2}

, then

30\beta^{3}

is equal to _______.

Q8·MathematicsSingle correctJEE Main 2026

The area of the region bounded by the curves

x + 3y^{2} = 0

and

x + 4y^{2} = 1

is equal to:

(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{4}{3}$
(D) $\frac{5}{3}$

Q9·MathematicsSingle correctJEE Main 2026

The area of the region

\{(x, y) : y \leq \pi - |x|,\ y \leq |x \sin x|,\ y \geq 0\}

is:

(A) $1 + \frac{\pi^2}{8}$
(B) $2 + \frac{\pi^2}{4}$
(C) $\frac{\pi^2}{8} - 1$
(D) $4 + \frac{\pi^2}{2}$

Q10·MathematicsNumericalJEE Main 2026

If the area of the region bounded by

16x^2 - 9y^2 = 144

and

8x - 3y = 24

is A, then

3(A + 6\log_e(3))

is equal to ________.

Q11·MathematicsSingle correctJEE Main 2026

The area of the region

R = \{(x, y) : xy \le 8, 1 \le y \le x^2, x \ge 0\}

is

(A) $\dfrac{1}{3}\left(49\log_e(2) - 15\right)$
(B) $\dfrac{2}{3}\left(20\log_e(2) + 9\right)$
(C) $\dfrac{2}{3}\left(24\log_e(2) - 7\right)$
(D) $\dfrac{1}{3}\left(40\log_e(2) + 27\right)$

Q12·MathematicsSingle correctJEE Main 2026

Let

P_{1} : y = 4x^{2}

and

P_{2} : y = x^{2} + 27

be two parabolas. If the area of the bounded region enclosed between

P_{1}

and

P_{2}

is six times the area of the bounded region enclosed between the line

y = \alpha x

, α > 0 and

P_{1}

, then α is equal to :

(A)8
(B)15
(C)12
(D)6

Q13·MathematicsSingle correctJEE Main 2026

Let

\mathrm{A}_{1}

be the bounded area enclosed by the curves

y = x^{2} + 2

,

x + y = 8

and y-axis that lies in the first quadrant. Let

\mathrm{A}_{2}

be the bounded area enclosed by the curves

y = x^{2} + 2

,

y^{2} = x

,

x = 2

, and y-axis that lies in the first quadrant. Then

\mathrm{A}_{1} - \mathrm{A}_{2}

is equal to

(A) $\frac{2}{3}\left(2\sqrt{2} + 1\right)$
(B) $\frac{2}{3}\left(4\sqrt{2} + 1\right)$
(C) $\frac{2}{3}\left(\sqrt{2} + 1\right)$
(D) $\frac{2}{3}\left(3\sqrt{2} + 1\right)$

Q14·MathematicsSingle correctJEE Main 2026

Let

f(\alpha)

denote the area of the region in the first quadrant bounded by

x = 0

,

x = 1

,

y^{2} = x

and

y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^{2}

. Then

(f(0) + f(1))

is equal to

(A)9
(B)14
(C)7
(D)12

Q15·MathematicsNumericalJEE Main 2026

Let the area of the region bounded by the curve

y = \max\ \{\sin x, \cos x\}

, lines

x = 0

,

x = \frac{3\pi}{2}

, and the x-axis be A. Then,

A + A^{2}

is equal to

Q16·MathematicsSingle correctJEE Main 2026

The area of the region enclosed between the circles

x^2 + y^2 = 4

and

x^2 + (y-2)^2 = 4

is :

(A) $\dfrac{2}{3}(2\pi - 3\sqrt{3})$
(B) $\dfrac{4}{3}(2\pi - 3\sqrt{3})$
(C) $\dfrac{4}{3}(2\pi - \sqrt{3})$
(D) $\dfrac{2}{3}(4\pi - 3\sqrt{3})$

Q17·MathematicsSingle correctJEE Main 2026

The area of the region

A = \{(x,\ y) : 4x^{2} + y^{2} \le 8 \text{ and } y^{2} \le 4x\}

is :

(A) $\frac{\pi}{2} + 2$
(B) $\pi + \frac{2}{3}$
(C) $\pi + 4$
(D) $\frac{\pi}{2} + \frac{1}{3}$

Q18·MathematicsSingle correctJEE Main 2026

Let the line x = −1 divide the area of the region

\left\{(x,y) : 1 + x^2 \le y \le 3 - x\right\}

in the ratio m : n, gcd (m, n) = 1. Then m + n is equal to

(A)25
(B)28
(C)26
(D)27

Q19·MathematicsSingle correctJEE Main 2026

If the area of the region

\{(x, y) : 1 - 2x \le y \le 4 - x^2, x \ge 0, y \ge 0\}

is

\frac{\alpha}{\beta}

, α, β, ∈ N, gcd (α, β) = 1, then the value of (α + β) is :

(A)73
(B)85
(C)91
(D)67

Q20·MathematicsSingle correctJEE Main 2026

The area of the region, inside the ellipse

x^{2} + 4y^{2} = 4

and outside the region bounded by the curves

y = |x| - 1

and

y = 1 - |x|

, is :

(A) $2(\pi - 1)$
(B) $2\pi - \frac{1}{2}$
(C) $3(\pi - 1)$
(D) $2\pi - 1$

Q21·MathematicsSingle correctJEE Advanced 2025

Let R denote the set of all real numbers. Then the area of the region

\left\{ (x, y) \in R \times R : x > 0,\, y > \frac{1}{x},\, 5x - 4y - 1 > 0,\, 4x + 4y - 17 < 0 \right\}

is

(A) $\frac{17}{16} - \log_e 4$
(B) $\frac{33}{8} - \log_e 4$
(C) $\frac{57}{8} - \log_e 4$
(D) $\frac{17}{2} - \log_e 4$

Q22·MathematicsMultiple correctJEE Advanced 2025

Let S denote the locus of mid-points of those chords of the parabola

y^2 = x

, such that area of the region enclosed between the parabola and the chord is

\frac{4}{3}

. Let R denote the region lying in the first quadrant, enclosed by the parabola

y^2 = x

, the curve S , and the lines x = 1 and x = 4. Then which of the following statements is (are) TRUE?

(A) $\left( 4, \sqrt{3} \right) \in S$
(B) $\left( 5, \sqrt{2} \right) \in S$
(C)Area of R is $\frac{14}{3} - 2\sqrt{3}$
(D)Area of R is $\frac{14}{3} - \sqrt{3}$

Q23·MathematicsIntegerJEE Main 2025

Let the area of the bounded region

\{(x,y):0\le9x\le y^2,\ y\ge3x-6\}

be

A

. Then

6A

is equal to ___

Q24·MathematicsSingle correctJEE Main 2025

If the area of the region

\{(x,y):1+x^2\le y\le \min\{x+7,\ 11-3x\}\}

is

A

, then

3A

is equal to:

(A)50
(B)49
(C)46
(D)47

Q25·MathematicsSingle correctJEE Main 2025

If the area of the region bounded by the curves

y=4-\dfrac{x^2}{4}

and

y=\dfrac{x-4}{2}

is equal to

\alpha

, then

6\alpha

equals

(A)250
(B)210
(C)240
(D)220

Q26·MathematicsIntegerJEE Main 2025

If the area of the region

\{(x,y):|x-5|\le y\le4\sqrt{x}\}

is

A

, then

3A

is equal to ______.

Q27·MathematicsSingle correctJEE Main 2025

Let

f:[0,\infty)\to\mathbb{R}

be differentiable function such that

f(x)=1-2x+\displaystyle\int_0^x e^{x-t}f(t)\,dt

for all

x\in[0,\infty)

. Then the area of the region bounded by

y=f(x)

and the coordinate axes is

(A) $\sqrt{5}$
(B) $\dfrac{1}{2}$
(C) $\sqrt{2}$
(D)2

Q28·MathematicsIntegerJEE Main 2025

The area of the region bounded by the curve

y=\max\{|x|,\,x|x-2|\}

, the x-axis and the lines

x=-2

and

x=4

is equal to __________.

Q29·MathematicsSingle correctJEE Main 2025

The area of the region

\{(x,y):|x-y|\le y\le 4\sqrt{x}\}

is:

(A)512
(B) $\dfrac{1024}{3}$
(C) $\dfrac{512}{3}$
(D) $\dfrac{2048}{3}$

Q30·MathematicsIntegerJEE Main 2025

If the area of the region

\{(x,y):|4-x^2|\le y\le x^2,\ y\le4,\ x\ge0\}

is

\left(\dfrac{80\sqrt{2}}{\alpha}-\beta\right)

,

\alpha,\beta\in N

, then

\alpha+\beta

is equal to ______.

Q31·MathematicsSingle correctJEE Main 2025

Let the area enclosed between the curves

|y|=1-x^2

and

x^2+y^2=1

be

\alpha

. If

9\alpha=\beta\pi+\gamma

;

\beta,\gamma

are integers, then the value of

|\beta-\gamma|

equals

(A)27
(B)18
(C)15
(D)33

Q32·MathematicsSingle correctJEE Main 2025

Let the area of the region

\{(x,y): 2y\le x^2+3,\ y+|x|\le 3,\ y\ge|x-1|\}

be A. Then 6A is equal to:

(A)16
(B)12
(C)18
(D)14

Q33·MathematicsSingle correctJEE Main 2025

The area of the region bounded by the curves

x(1+y^2)=1

and

y^2=2x

is:

(A) $2\left(\frac{\pi}{2}-\frac{1}{3}\right)$
(B) $\frac{\pi}{4}-\frac{1}{3}$
(C) $\frac{\pi}{2}-\frac{1}{3}$
(D) $\frac{1}{2}\left(\frac{\pi}{2}-\frac{1}{3}\right)$

Q34·MathematicsSingle correctJEE Main 2025

The area (in sq. units) of the region

\{(x, y): 0\le y\le 2|x|+1, 0\le y\le x^2+1, |x|\le 3\}

is

(A) $\frac{80}{3}$
(B) $\frac{64}{3}$
(C) $\frac{17}{3}$
(D) $\frac{32}{3}$

Q35·MathematicsSingle correctJEE Main 2025

The area of the region

\{(x,y):x^2+4x+2\le y\le|x+2|\}

is equal to

(A) $7$
(B) $\dfrac{24}{5}$
(C) $\dfrac{20}{3}$
(D) $5$

Q36·MathematicsSingle correctJEE Main 2025

The area of the region enclosed by the curves

y=e^x

,

y=|e^x-1|

and y-axis is:

(A) $1+\log_e 2$
(B) $2\log_e 2$
(C) $2\log_e 2-1$
(D) $1-\log_e 2$

Q37·MathematicsSingle correctJEE Main 2025

If the area of the region

\{(x,y):-1\le x\le 1,\ 0\le y\le a+e^{|x|}-e^{-x},\ a>0\}

is

\dfrac{e^2+8e+1}{e}

, then the value of

a

is :

(A)7
(B)6
(C)8
(D)5

Q38·MathematicsSingle correctJEE Main 2025

The area of the region inside the circle

(x-2\sqrt3)^2+y^2=12

and outside the parabola

y^2=2\sqrt3\,x

is:

(A) $6\pi-8$
(B) $3\pi-8$
(C) $6\pi-16$
(D) $5\pi+8$

Q39·MathematicsSingle correctJEE Main 2025

Let

f:\mathbb{R}\to\mathbb{R}

be twice differentiable with

f(x+y)=f(x)f(y)

for all

x,y

. If

f'(0)=4a

and

f''(x)-3a\,f'(x)-f(x)=0\,(a>0)

, then the area of

R=\{(x,y):0\le y\le f(ax),\,0\le x\le 2\}

is:

(A) $e^{4}-1$
(B) $e^{2}+1$
(C) $e^{4}+1$
(D) $e^{2}-1$

Q40·MathematicsSingle correctJEE Main 2025

The area of the region enclosed by the curves

y=x^2-4x+4

and

y^2=16-8x

is:

(A) $\dfrac{8}{3}$
(B) $\dfrac{4}{3}$
(C)5
(D)8

Q41·MathematicsSingle correctJEE Advanced 2024

Let

S = \left\{(x, y) \in \mathbb{R} \times \mathbb{R} : x \geq 0, y \geq 0, y^2 \leq 4x, y^2 \leq 12 - 2x \text{ and } 3y + \sqrt{8}x \leq 5\sqrt{8}\right\}

. If the area of the region

S

is

\alpha\sqrt{2}

, then

\alpha

is equal to

(A) $\frac{17}{2}$
(B) $\frac{17}{3}$
(C) $\frac{17}{4}$
(D) $\frac{17}{5}$

Q42·MathematicsSingle correctJEE Main 2024

The area (in square units) of the region enclosed by the ellipse

x^{2}+3y^{2}=18

in the first quadrant below the line

y=x

is:

(A) $\sqrt{3}\pi+\dfrac{3}{4}$
(B) $\sqrt{3}\pi$
(C) $\sqrt{3}\pi-\dfrac{3}{4}$
(D) $\sqrt{3}\pi+1$

Q43·MathematicsSingle correctJEE Main 2024

The parabola

y^2 = 4x

divides the area of the circle

x^2 + y^2 = 5

in two parts. The area of the smaller part is equal to:

(A) $\frac{2}{3} + 5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
(B) $\frac{1}{3} + 5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
(C) $\frac{1}{3} + \sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$
(D) $\frac{2}{3} + \sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)$

Q44·MathematicsSingle correctJEE Main 2024

The area of the region in the first quadrant inside the circle

x^2+y^2=8

and outside the parabola

y^2=2x

is equal to

(A) $\tfrac{\pi}{2}-\tfrac{1}{3}$
(B) $\pi-\tfrac{2}{3}$
(C) $\tfrac{\pi}{2}-\tfrac{2}{3}$
(D) $\pi-\tfrac{1}{3}$

Q45·MathematicsNumericalJEE Main 2024

Let the area of the region enclosed by the curve

y=\min\{\sin x,\cos x\}

and the x-axis between

x=-\pi

to

x=\pi

be

A

. Then

A^2

is equal to ___

Q46·MathematicsSingle correctJEE Main 2024

If the area of the region

\left\{(x,y):\frac{a}{x^{2}}\le y\le\frac{1}{x},\,1\le x\le 2,\,0<a<1\right\}

is

(\log_{e}2)-\frac{1}{7}

then the value of

7a-3

is equal to:

(A) $2$
(B) $0$
(C) $-1$
(D) $1$

Q47·MathematicsSingle correctJEE Main 2024

Let the area of the region enclosed by the curves

y=3x

,

2y=27-3x

and

y=3x-x\sqrt{x}

be

A

. Then

10A

is equal to

(A) $184$
(B) $154$
(C) $172$
(D) $162$

Q48·MathematicsSingle correctJEE Main 2024

The area enclosed between the curves

y=x|x|

and

y=x-|x|

is:

(A) $\dfrac{8}{3}$
(B) $\dfrac{2}{3}$
(C)1
(D) $\dfrac{4}{3}$

Q49·MathematicsNumericalJEE Main 2024

The area of the region enclosed by the parabolas

y = x^2 - 5x

and

y = 7x - x^2

is ___.

Q50·MathematicsSingle correctJEE Main 2024

The area (in sq. units) of the region described by

\{(x,y):y^2\le 2x \text{ and } y\ge 4x-1\}

is

(A) $\tfrac{11}{32}$
(B) $\tfrac{8}{3}$
(C) $\tfrac{11}{12}$
(D) $\tfrac{9}{32}$

Q51·MathematicsSingle correctJEE Main 2024

One of the points of intersection of the curves

y=1+3x-2x^2

and

y=\dfrac1x

is

\left(\dfrac12,2\right)

. Let the area of the region bounded by these curves be

\dfrac{1}{24}(\ell\sqrt5+m)-n\log_e(1+\sqrt5)

, where

\ell,m,n\in\mathbb{N}

. Then

\ell+m+n

is equal to:

(A) $32$
(B) $30$
(C) $29$
(D) $31$

Q52·MathematicsSingle correctJEE Main 2024

The area (in sq. units) of the region

S=\{z=x+iy:|z-1|\le 2,\;(z+\bar{z})+i(z-\bar{z})\le 2,\;\operatorname{Im}(z)\ge 0\}

is

(A) $\tfrac{3\pi}{2}$
(B) $3\pi$
(C) $\tfrac{17\pi}{6}$
(D) $\tfrac{7\pi}{4}$

Q53·MathematicsSingle correctJEE Main 2024

The area enclosed by the curves

xy+4y=16

and

x+y=6

is equal to:

(A) $28-30\log_e 2$
(B) $30-28\log_e 2$
(C) $30-32\log_e 2$
(D) $32-30\log_e 2$

Q54·MathematicsNumericalJEE Main 2024

The sum of squares of all possible values of

k

, for which area of the region bounded by the parabolas

2y^2=kx

and

ky^2=2(y-x)

is maximum, is equal to __________.

Q55·MathematicsNumericalJEE Main 2024

Three points

O(0,0)

,

P(a,a^2)

,

Q(-b,b^2)

,

a>0,b>0

, are on the parabola

y=x^2

. Let

S_1

be the area of the region bounded by the line PQ and the parabola, and

S_2

be the area of the triangle OPQ. If the minimum value of

\dfrac{S_1}{S_2}

is

\dfrac{m}{n}

,

\gcd(m,n)=1

, then

m+n

is equal to __________.

Q56·MathematicsSingle correctJEE Main 2024

The area of the region

\left\{(x,y): y^2\le 4x,\ x<4,\ \dfrac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,\ x\ne 3\right\}

is

(A) $\dfrac{16}{3}$
(B) $\dfrac{64}{3}$
(C) $\dfrac{8}{3}$
(D) $\dfrac{32}{3}$

Q57·MathematicsSingle correctJEE Main 2024

The area of the region enclosed by the parabola

y=4x-x^2

and

3y=(x-4)^2

is equal to

(A) $\dfrac{32}{9}$
(B)4
(C)6
(D) $\dfrac{14}{3}$

Q58·MathematicsNumericalJEE Main 2024

The area of the region enclosed by the parabola

(y-2)^2=x-1

, the line

x-2y+4=0

and the positive coordinate axes is ______.

Q59·MathematicsSingle correctJEE Main 2024

The area (in square units) of the region bounded by the parabola

y^2=4(x-2)

and the line

y=2x-8

is:

(A) $8$
(B) $9$
(C) $6$
(D) $7$

Q60·MathematicsNumericalJEE Main 2024

Let the area of the region

\{(x,y):0\le x\le 3,\ 0\le y\le\min\{x^2+2,\,2x+2\}\}

be

A

. Then

12A

is equal to ___.

Q61·MathematicsNumericalJEE Main 2024

The area (in sq. units) of the part of circle

x^2+y^2=169

which is below the line

5x-y=13

is

\dfrac{\pi\alpha}{2\beta}-\dfrac{65}{2}+\dfrac{\alpha}{\beta}\sin^{-1}\left(\dfrac{12}{13}\right)

where

\alpha,\beta

are coprime numbers. Then

\alpha+\beta

is equal to ______.

Q62·MathematicsNumericalJEE Main 2024

If the area of the region

\{(x,y):0\le y\le\min\{2x,6x-x^2\}\}

is A, then

12A

is equal to __________.

Q63·MathematicsNumericalJEE Main 2024

Let the area of the region

\{(x,y):x-2y+4\ge 0,\,x+2y^2\ge 0,\,x+4y^2\le 8,\,y\ge 0\}

be

\dfrac{m}{n}

, where

m

and

n

are coprime numbers. Then

m+n

is equal to __________.

Q64·MathematicsNumericalJEE Advanced 2023

Let

n \geq 2

be a natural number and

f : [0, 1] \rightarrow R

be the function defined by

f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \leq x \leq \frac{1}{2n} \\ 2n(2nx - 1) & \text{if } \frac{1}{2n} \leq x \leq \frac{3}{4n} \\ 4n(1 - nx) & \text{if } \frac{3}{4n} \leq x \leq \frac{1}{n} \\ \frac{n}{n-1}(nx - 1) & \text{if } \frac{1}{n} \leq x \leq 1 \end{cases}

If n is such that the area of the region bounded by the curves

x = 0

,

x = 1

,

y = 0

and

y = f(x)

is 4 , then the maximum value of the function f is

Q65·MathematicsMultiple correctJEE Advanced 2023

Let

f : [0, 1] \rightarrow [0, 1]

be the function defined by

f(x) = \frac{x^3}{3} - x^2 + \frac{5}{9}x + \frac{17}{36}

. Consider the square region

S = [0, 1] \times [0, 1]

. Let

G = \{(x, y) \in S : y > f(x)\}

be called the green region and

R = \{(x, y) \in S : y < f(x)\}

be called the red region. Let

L_h = \{(x, h) \in S : x \in [0, 1]\}

be the horizontal line drawn at a height

h \in [0, 1]

. Then which of the following statements is(are) true?

(A)There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the green region below the line $L_h$
(B)There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the red region below the line $L_h$
(C)There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the red region below the line $L_h$
(D)There exists an $h \in \left[\frac{1}{4}, \frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the green region below the line $L_h$

Q66·MathematicsNumericalJEE Main 2023

If the area bounded by the curve

2y^2 = 3x

, lines

x + y = 3

,

y = 0

and outside the circle

(x - 3)^2 + y^2 = 2

is

A

, then

4(\pi + 4A)

is equal to____

Q67·MathematicsSingle correctJEE Main 2023

The area of the region enclosed by the curve

f(x)=\max\{\sin x,\cos x\},\,-\pi\le x\le\pi

and the

x

-axis is:

(A) $2(\sqrt2+1)$
(B) $2\sqrt2(\sqrt2+1)$
(C) $4(\sqrt2)$
(D) $4$

Q68·MathematicsSingle correctJEE Main 2023

The area of the region

\left\{(x, y) : x^2 \le y \le \left|x^2 - 4\right|,\ y \ge 1\right\}

is

(A) $\dfrac{3}{4}\left(4\sqrt{2} - 1\right)$
(B) $\dfrac{4}{3}\left(4\sqrt{2} - 1\right)$
(C) $\dfrac{4}{3}\left(4\sqrt{2} + 1\right)$
(D) $\dfrac{3}{4}\left(4\sqrt{2} + 1\right)$

Q69·MathematicsSingle correctJEE Main 2023

The area of the region enclosed by the curve

y=x^3

and its tangent at the point

(-1,-1)

is

(A) $\dfrac{27}{4}$
(B) $\dfrac{19}{4}$
(C) $\dfrac{23}{4}$
(D) $\dfrac{31}{4}$

Q70·MathematicsNumericalJEE Main 2023

If

A

is the area in the first quadrant enclosed by the curve

C:2x^2-y+1=0

, the tangent to

C

at the point

(1,3)

and the line

x+y=1

, then the value of

60A

is

Q71·MathematicsSingle correctJEE Main 2023

Area of the region

\{ (x, y) : x^2 + (y - 2)^2 \le 4,\ x^2 \ge 2y \}

is

(A) $2\pi - \frac{16}{3}$
(B) $\pi - \frac{8}{3}$
(C) $\pi + \frac{8}{3}$
(D) $2\pi + \frac{16}{3}$

Q72·MathematicsNumericalJEE Main 2023

If the area of the region

\{(x,y):|x^2-2|\le y\le x\}

is A, then

6A+16\sqrt{2}

is equal to _______ .

Q73·MathematicsNumericalJEE Main 2023

Let

y=p(x)

be the parabola passing through the points

(-1,0),(0,1)

and

(1,0)

. If the area of the region

\{(x,y):(x+1)^2+(y-1)^2\le1,\ y\le p(x)\}

is A, then

12(\pi-4A)

is equal to _________.

Q74·MathematicsSingle correctJEE Main 2023

The area of the region

\{(x,y):x^{2}\le y\le 8-x^{2},\, y\le 7\}

is

(A)21
(B)18
(C)24
(D)20

Q75·MathematicsNumericalJEE Main 2023

Let the area enclosed by the lines

x+y=2

,

y=0

,

x=0

and the curve

f(x)=\min\left\{x^{2}+\dfrac{3}{4},\,1+[x]\right\}

, where

[x]

denotes the greatest integer

\leq x

, be A. Then the value of

12A

is

Q76·MathematicsSingle correctJEE Main 2023

The area bounded by the curves

y=|x-1|+|x-2|

and

y=3

is equal to

(A)3
(B)4
(C)5
(D)6

Q77·MathematicsNumericalJEE Main 2023

If the area of the region

S=\{(x,y):2y-y^{2}\le x^{2}\le 2y,\,x\ge y\}

is equal to

\dfrac{n+2}{n+1}-\dfrac{\pi}{n-1}

, then the natural number

n

is equal to _____.

Q78·MathematicsNumericalJEE Main 2023

Let

A

be the area bounded by the curve

y=x|x-3|

, the

x

-axis and the ordinates

x=-1

and

x=2

. Then

12A

is equal to _______ .

Q79·MathematicsSingle correctJEE Main 2023

The area of the region given by

\{(x,y):xy\leq 8,\,1\leq y\leq x^2\}

is:

(A) $16\log_e 2+\dfrac73$
(B) $16\log_e 2-\dfrac{14}{3}$
(C) $8\log_e 2-\dfrac{13}{3}$
(D) $8\log_e 2+\dfrac76$

Q80·MathematicsNumericalJEE Main 2023

Let the area of the region

\{(x,y):|2x-1|\le y\le|x^2-x|,\ 0\le x\le1\}

be

A

. Then

(6A+11)^2

is equal to

Q81·MathematicsSingle correctJEE Main 2023

Let

q

be the maximum integral value of

p

in

[0, 10]

for which the roots of the equation

x^2 - px + \dfrac{5}{4}p=0

are rational. Then the area of the region

\{(x, y): 0 \le y \le (x - q)^2,\ 0 \le x \le q\}

is:

(A) $243$
(B) $164$
(C) $\dfrac{125}{3}$
(D) $25$

Q82·MathematicsNumericalJEE Main 2023

Let

A

be the area of the region

\{(x, y): y \ge x^2,\ y \ge (1 - x)^2,\ y \le 2x(1 - x)\}

. Then

540\,A

is equal to _______.

Q83·MathematicsNumericalJEE Main 2023

Let

\alpha

be the area of the larger region bounded by the curve

y^2=8x

and the lines

y=x

and

x=2

, which lies in the first quadrant. Then the value of

3\alpha

is equal to

Q84·MathematicsSingle correctJEE Main 2023

The area of the region

A=\{(x,y):|\cos x-\sin x|\le y\le\sin x,\,0\le x\le\dfrac{\pi}{2}\}

is:

(A) $\sqrt5+2\sqrt2-4.5$
(B) $1-\dfrac{3}{\sqrt2}+\dfrac{4}{\sqrt5}$
(C) $\dfrac{3}{\sqrt5}-\dfrac{3}{\sqrt2}+1$
(D) $\sqrt5-2\sqrt2+1$

Q85·MathematicsSingle correctJEE Main 2023

Let

\Delta

be the area of the region

\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}

. Then

\dfrac{1}{2}\left(\Delta - 21\sin^{-1}\dfrac{2}{\sqrt{7}}\right)

is equal to

(A) $2\sqrt{3} - \dfrac{2}{3}$
(B) $\sqrt{3} - \dfrac{4}{3}$
(C) $\sqrt{3} - \dfrac{2}{3}$
(D) $2\sqrt{3} - \dfrac{1}{3}$

Q86·MathematicsSingle correctJEE Main 2023

Let

A = \{(x, y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4 - (x - 1)^2}\}

and

B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : 0 \leq y \leq \min\{2x, \sqrt{4 - (x - 1)^2}\}\}

. Then the ratio of the area of

A

to the area of

B

is

(A) $\dfrac{\pi + 1}{\pi - 1}$
(B) $\dfrac{\pi}{\pi - 1}$
(C) $\dfrac{\pi - 1}{\pi + 1}$
(D) $\dfrac{\pi}{\pi + 1}$

Q87·MathematicsNumericalJEE Main 2023

If the area enclosed by the parabolas

P_1:2y=5x^2

and

P_2:x^2-y+6=0

is equal to the area enclosed by

P_1

and

y=\alpha x

,

\alpha>0

, then

\alpha^3

is equal to _______.

Q88·MathematicsNumericalJEE Main 2023

If the area of the region bounded by the curves

y^2-2y=-x,\ x+y=0

is A, then

8A

is equal to

Q89·MathematicsSingle correctJEE Main 2023

The area enclosed by the curves

y^2+4x=4

and

y-2x=2

is:

(A) $9$
(B) $\tfrac{22}{3}$
(C) $\tfrac{23}{3}$
(D) $\tfrac{25}{3}$

Q90·MathematicsIntegerJEE Advanced 2022

Consider the function f, g : R → R defined by

f\left(x\right) = x^{2} + \frac{5}{12}

and

g(x) = \begin{cases} 2\left(1 - \frac{4|x|}{3}\right), & |x| \leq \frac{3}{4}, \\ 0, & |x| > \frac{3}{4}. \end{cases}

If α is the area of the region

\left\{\left(x, y\right) \in R \times R : |x| \leq \frac{3}{4}, 0 \leq y \leq \min\left\{f\left(x\right), g\left(x\right)\right\}\right\}

, then the value of 9α is __________.

Q91·MathematicsSingle correctJEE Main 2022

The area of the region

\left\{(x,\,y) : |\,x-1\,| \le y \le \sqrt{5-x^{2}}\right\}

is equal to :

(A) $\frac{5}{2}\sin^{-1}\left(\frac{3}{5}\right)-\frac{1}{2}$
(B) $\frac{5\pi}{4}-\frac{3}{2}$
(C) $\frac{3\pi}{4}+\frac{3}{2}$
(D) $\frac{5\pi}{4}-\frac{1}{2}$

Q92·MathematicsSingle correctJEE Main 2022

The area enclosed by the curves

y=\log_{e}\left(x+e^{2}\right)

,

x=\log_{e}\left(\frac{2}{y}\right)

and

x=\log_{e}2

, above the line

y=1

is

(A) $2+e-\log_{e}2$
(B) $1+e-\log_{e}2$
(C) $e-\log_{e}2$
(D) $1+\log_{e}2$

Q93·MathematicsSingle correctJEE Main 2022

The area of the smaller region enclosed by the curves

y^2 = 8x + 4

and

x^2 + y^2 + 4\sqrt{3}x - 4 = 0

is equal to

(A) $\frac{1}{3}\left(2 - 12\sqrt{3} + 8\pi\right)$
(B) $\frac{1}{3}\left(2 - 12\sqrt{3} + 6\pi\right)$
(C) $\frac{1}{3}\left(4 - 12\sqrt{3} + 8\pi\right)$
(D) $\frac{1}{3}\left(4 - 12\sqrt{3} + 6\pi\right)$

Q94·MathematicsSingle correctJEE Main 2022

The area of the region enclosed by

y \le 4x^{2}, x^{2} \le 9y

and

y \le 4

, is equal to :

(A) $\frac{40}{3}$
(B) $\frac{56}{3}$
(C) $\frac{112}{3}$
(D) $\frac{80}{3}$

Q95·MathematicsSingle correctJEE Main 2022

The odd natural number a, such that the area of the region bounded by y= 1, y = 3, x = 0, x = y

^{a}

is

\dfrac{364}{3}

, equal to :

(A)3
(B)5
(C)7
(D)9

Q96·MathematicsSingle correctJEE Main 2022

The area bounded by the curves

y=|x^{2}-1|

and

y=1

is

(A) $\frac{2}{3}\left(\sqrt{2}+1\right)$
(B) $\frac{4}{3}\left(\sqrt{2}-1\right)$
(C) $2\left(\sqrt{2}-1\right)$
(D) $\frac{8}{3}\left(\sqrt{2}-1\right)$

Q97·MathematicsSingle correctJEE Main 2022

The area of the region given by

A = \{(x, y) : x^{2} \le y \le \min \{x + 2, 4 - 3x\}\}

is :

(A) $\frac{31}{8}$
(B) $\frac{17}{6}$
(C) $\frac{19}{6}$
(D) $\frac{27}{8}$

Q98·MathematicsSingle correctJEE Main 2022

The area enclosed by

y^{2} = 8x

and

y = \sqrt{2}\,x

that lies outside the triangle formed by

y = \sqrt{2}x

,

x = 1

,

y = 2\sqrt{2}

, is equal to :

(A) $\frac{16\sqrt{2}}{6}$
(B) $\frac{11\sqrt{2}}{6}$
(C) $\frac{13\sqrt{2}}{6}$
(D) $\frac{5\sqrt{2}}{6}$

Q99·MathematicsNumericalJEE Main 2022

For real numbers

a

, b (

a > b > 0

), let

Area\left\{(x,y) : x^2 + y^2 \leq a^2 \text{and} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1\right\} = 30\pi

and

Area\left\{(x,y) : x^2 + y^2 \geq b^2 \text{and} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\right\} = 18\pi

Then the value of

(a - b)^2

is equal to____.

Q100·MathematicsSingle correctJEE Main 2022

The area of the region

\mathrm{S}=\{(x,y):y^{2} \le 8x, y \ge \sqrt{2}x, x \ge 1\}

is

(A) $\frac{13\sqrt{2}}{6}$
(B) $\frac{11\sqrt{2}}{6}$
(C) $\frac{5\sqrt{2}}{6}$
(D) $\frac{19\sqrt{2}}{6}$

Q101·MathematicsSingle correctJEE Main 2022

The area of the bounded region enclosed by the curve

y = 3 - \left|x - \frac{1}{2}\right| - |x + 1|

and the x-axis is

(A) $\frac{9}{4}$
(B) $\frac{45}{16}$
(C) $\frac{27}{8}$
(D) $\frac{63}{16}$

Q102·MathematicsNumericalJEE Main 2022

If the area of the region

\left\{(x,y):x^{\frac{2}{3}}+y^{\frac{2}{3}}\le 1\, x+y\ge 0,\ y\ge 0\right\}

is A, then

\dfrac{256\,A}{\pi}

is

Q103·MathematicsNumericalJEE Main 2022

Let

A_1=\left\{(x,y):|x|\le y^2, |x|+2y\le 8\right\}

and

A_2=\left\{(x,y):|x|+|y|\le k\right\}

. If 27 (Area

A_1

) = 5 (Area

A_2

), then k is equal to :

Q104·MathematicsSingle correctJEE Main 2022

The area of the region bounded by

y^2=8x

and

y^2=16(3-x)

is equal to :-

(A) $\frac{32}{3}$
(B) $\frac{40}{3}$
(C)16
(D)19

Q105·MathematicsSingle correctJEE Main 2022

The area of the region enclosed between the parabolas

y^2 = 2x - 1

and

y^2 = 4x - 3

is

(A) $\frac{1}{3}$
(B) $\frac{1}{6}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$

Q106·MathematicsNumericalJEE Main 2022

Let S be the region bounded by the curves

y=x^{3}

and

y^{2}=x

. The curve

y=2|x|

divides S into two regions of areas

R_{1}

and

R_{2}

. If max

\{R_{1},R_{2}\}=R_{2}

, then

\dfrac{R_{2}}{R_{1}}

is equal to ____ .

Q107·MathematicsNumericalJEE Main 2022

The area (in sq. units) of the region enclosed between the parabola

y^{2} = 2x

and the line

x + y = 4

is ______.

Q108·MathematicsSingle correctJEE Advanced 2021

The area of the region

\left\{(x, y) : 0 \le x \le \frac{9}{4},\ 0 \le y \le 1,\ x \ge 3y,\ x + y \ge 2\right\}

is

(A) $\frac{11}{32}$
(B) $\frac{35}{96}$
(C) $\frac{37}{96}$
(D) $\frac{13}{32}$

Q109·MathematicsSingle correctJEE Main 2021

The area, enclosed by the curves y = sin x + cos x and

y = \left|\cos x - \sin x\right|

and the lines x = 0,

x = \frac{\pi}{2}

, is :

(A) $2\sqrt{2}\left(\sqrt{2} - 1\right)$
(B) $2\left(\sqrt{2} + 1\right)$
(C) $4\left(\sqrt{2} - 1\right)$
(D) $2\sqrt{2}\left(\sqrt{2} + 1\right)$

Q110·MathematicsNumericalJEE Main 2021

If the line

y = mx

bisects the area enclosed by the lines

x = 0

,

y = 0

,

x = \frac{3}{2}

and the curve

y = 1 + 4x - x^{2}

, then 12 m is equal to ______.

Q111·MathematicsSingle correctJEE Main 2021

The area of the region bounded by the parabola

(y - 2)^{2} = (x - 1)

, the tangent to it at the point whose ordinate is 3 and the x-axis is :

(A)9
(B)10
(C)4
(D)6

Q112·MathematicsNumericalJEE Main 2021

The area of the region

S = \{(x, y) : 3x^{2} \leq 4y \leq 6x + 24\}

is ________.

Q113·MathematicsNumericalJEE Main 2021

Let a and b respectively be the points of local maximum and local minimum of the function

f(x) = 2x^{3} - 3x^{2} - 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 ________.

Q114·MathematicsSingle correctJEE Main 2021

If the area of the bounded region

R = \left\{ (x, y) : \max\left\{0, \log_e x\right\} \le y \le 2^x, \frac{1}{2} \le x \le 2 \right\}

is,

\ \alpha\left(\log_e 2\right)^{-1} + \beta\left(\log_e 2\right) + \gamma

, then the value of

\left(\alpha + \beta - 2\gamma\right)^2

is equal to :

(A)4
(B)8
(C)2
(D)1

Q115·MathematicsSingle correctJEE Main 2021

The area of the region bounded by y − x = 2 and x

^2

= y is equal to :

(A) $\frac{2}{3}$
(B) $\frac{9}{2}$
(C) $\frac{16}{3}$
(D) $\frac{4}{3}$

Q116·MathematicsSingle correctJEE Main 2021

The area (in sq. units) of the region, given by the set

\left\{(x, y) \in R \times R \mid x \geq 0, 2x^{2} \leq y \leq 4 - 2x\right\}

is :

(A) $\frac{17}{3}$
(B) $\frac{8}{3}$
(C) $\frac{7}{3}$
(D) $\frac{13}{3}$

Q117·MathematicsNumericalJEE Main 2021

The area (in sq. units) of the region bounded by the curves

x^2 + 2y - 1 = 0

,

y^2 + 4x - 4 = 0

and

y^2 - 4x - 4 = 0

, in the upper half plane is.........

Q118·MathematicsSingle correctJEE Main 2021

The area bounded by the curve

4y^{2} = x^{2}(4-x)(x-2)

is equal to :

(A) $\frac{\pi}{8}$
(B) $\frac{3\pi}{8}$
(C) $\frac{3\pi}{2}$
(D) $\frac{\pi}{16}$

Q119·MathematicsNumericalJEE Main 2021

Let

f : [-3, 1] \rightarrow R

be given as

f(x) = \begin{cases} \min\left\{(x+6), x^{2}\right\}, & -3 \le x \le 0 \\ \max\left\{\sqrt{x}, x^{2}\right\}, & 0 \le x \le 1. \end{cases}

If the area bounded by

y = f(x)

and x-axis is A, then the value of

6A

is equal to ______.

Q120·MathematicsNumericalJEE Main 2021

Let the curve y = y(x) be the solution of the differential equation,

\frac{dy}{dx} = 2(x+1)

. If the numerical value of area bounded by the curve y = y(x) and x-axis is

\frac{4\sqrt{8}}{3}

, then the value of y(1) is equal to _______ .

Q121·MathematicsSingle correctJEE Main 2021

Let

A_{1}

be the area of the region bounded by the curves y = sinx, y = cos x and y-axis in the first quadrant. Also, let

A_{2}

be the area of the region bounded by the curves y = sin x, y = cos x, x-axis and x=

\frac{\pi}{2}

in the first quadrant. Then,

(A) $A_{1} = A_{2}$ and $A_{1} + A_{2} = \sqrt{2}$
(B) $A_{1} : A_{2} = 1 : 2$ and $A_{1} + A_{2} = 1$
(C) $2A_{1} = A_{2}$ and $A_{1} + A_{2} = 1 + \sqrt{2}$
(D) $A_{1} : A_{2} = 1 : \sqrt{2}$ and $A_{1} + A_{2} = 1$

Q122·MathematicsNumericalJEE Main 2021

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

A^4

is equal to __________

Q123·MathematicsSingle correctJEE Main 2021

The area (in sq. units) of the part of the circle

x^2+y^2=36

, which is outside the parabola

y^2=9x

, is :

(A) $24\pi + 3\sqrt{3}$
(B) $12\pi + 3\sqrt{3}$
(C) $12\pi - 3\sqrt{3}$
(D) $24\pi - 3\sqrt{3}$

Q124·MathematicsSingle correctJEE Main 2021

The area of the region :

R=\left\{(x,\,y):5x^{2}\le y\le 2x^{2}+9\right\}

is:

(A) $9\sqrt{3}$ square units
(B) $12\sqrt{3}$ square units
(C) $11\sqrt{3}$ square units
(D) $6\sqrt{3}$ square units

Q125·MathematicsSingle correctJEE Advanced 2020

Let the functions

f : \mathbb{R} \to \mathbb{R}

and

g : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = e^{x-1} - e^{-|x-1|}

and

g(x) = \frac{1}{2}\left(e^{x-1} + e^{1-x}\right)

. Then the area of the region in the first quadrant bounded by the curves

y = f(x)

,

y = g(x)

and

x = 0

is

(A) $\left(2 - \sqrt{3}\right) + \frac{1}{2}\left(e - e^{-1}\right)$
(B) $\left(2 + \sqrt{3}\right) + \frac{1}{2}\left(e - e^{-1}\right)$
(C) $\left(2 - \sqrt{3}\right) + \frac{1}{2}\left(e + e^{-1}\right)$
(D) $\left(2 + \sqrt{3}\right) + \frac{1}{2}\left(e + e^{-1}\right)$

Q126·MathematicsSingle correctJEE Main 2020

The area (in sq. units) of the region enclosed by the curves

y = x^{2} - 1

and

y = 1 - x^{2}

is equal to

(A) $\frac{4}{3}$
(B) $\frac{8}{3}$
(C) $\frac{7}{2}$
(D) $\frac{16}{3}$

Q127·MathematicsSingle correctJEE Main 2020

The area (in sq. units) of the region

A = \{(x,y) : |x| + |y| \le 1,\ 2y^2 \ge |x|\}

is:

(A) $\dfrac{1}{3}$
(B) $\dfrac{7}{6}$
(C) $\dfrac{1}{6}$
(D) $\dfrac{5}{6}$

Q128·MathematicsSingle correctJEE Main 2020

The area (in sq. units) of the region

\{(x, y) : 0 \leq y \leq x^{2} + 1,\ 0 \leq y \leq x + 1,\ \frac{1}{2} \leq x \leq 2\}

is:

(A) $\frac{79}{16}$
(B) $\frac{23}{16}$
(C) $\frac{79}{24}$
(D) $\frac{23}{6}$

Q129·MathematicsSingle correctJEE Main 2020

Given :

f(x) = \begin{cases} x, & 0 \leq x < \dfrac{1}{2} \\[4pt] \dfrac{1}{2}, & x = \dfrac{1}{2} \\[4pt] 1 - x, & \dfrac{1}{2} < x \leq 1 \end{cases}

and

g(x) = \left(x - \dfrac{1}{2}\right)^{2}, x \in R

. Then the area (in sq. units) of the region bounded by the curves,

y = f(x)

and

y = g(x)

between the lines

2x = 1

and

2x = \sqrt{3}

, is:

(A) $\dfrac{1}{3} + \dfrac{\sqrt{3}}{4}$
(B) $\dfrac{1}{2} + \dfrac{\sqrt{3}}{4}$
(C) $\dfrac{\sqrt{3}}{4} - \dfrac{1}{3}$
(D) $\dfrac{1}{2} - \dfrac{\sqrt{3}}{4}$

Q130·MathematicsSingle correctJEE Main 2020

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) $\frac{1}{3}(12\pi - 1)$
(B) $\frac{1}{3}(6\pi - 1)$
(C) $\frac{1}{6}(12\pi - 1)$
(D) $\frac{1}{6}(24\pi - 1)$

Q131·MathematicsSingle correctJEE Main 2020

The area (in sq. units of the region

\{(x,y)\in R^{2}\,|\,4x^{2}\le y\le 8x+12\}

is:

(A) $\dfrac{124}{3}$
(B) $\dfrac{125}{3}$
(C) $\dfrac{128}{3}$
(D) $\dfrac{127}{3}$

Q132·MathematicsSingle correctJEE Advanced 2019

The area of the region

\{(x, y) : xy \le 8, 1 \le y \le x^2\}

is

(A) $16\log_e 2 - 6$
(B) $8\log_e 2 - \frac{14}{3}$
(C) $16\log_e 2 - \frac{14}{3}$
(D) $8\log_e 2 - \frac{7}{3}$

Q133·MathematicsSingle correctJEE Main 2019

If the area (in sq. units) bounded by the parabola

y^{2} = 4\lambda x

and the line

y = \lambda x

,

\lambda > 0

, is

\dfrac{1}{9}

, then

\lambda

is equal to :

(A)48
(B) $4\sqrt{3}$
(C) $2\sqrt{6}$
(D)24

Q134·MathematicsSingle correctJEE Main 2019

If the area (in sq. units) of the region

\{(x,y): y^{2} \le 4x, x+y \le 1, x \ge 0, y \ge 0\}

is

a\sqrt{2}+b

, then a

-

b is equal to :

(A) $\frac{10}{3}$
(B)6
(C) $\frac{8}{3}$
(D) $-\frac{2}{3}$

Q135·MathematicsSingle correctJEE Main 2019

The area(in sq. units) of the region bounded by the curves

y = 2^{x}

and

y = |x + 1|

in the first quadrant is

(A) $\dfrac{3}{2}$
(B) $\log_{e} 2 + \dfrac{3}{2}$
(C) $\dfrac{3}{2} - \dfrac{1}{\log_{e} 2}$
(D) $\dfrac{1}{2}$

Q136·MathematicsSingle correctJEE Main 2019

The area (in sq. units) of the region

A=\left\{\left(x,y\right):\dfrac{y^{2}}{2}\leq x\leq y+4\right\}

is:

(A) $\dfrac{53}{3}$
(B)18
(C)30
(D)16

Q137·MathematicsSingle correctJEE Main 2019

The area (in sq. units) of the region

A=\{(x, y) : x^{2}\leq y\leq x+2\}

is:

(A) $\dfrac{31}{6}$
(B) $\dfrac{13}{6}$
(C) $\dfrac{9}{2}$
(D) $\dfrac{10}{3}$

Q138·MathematicsSingle correctJEE Main 2019

The area (in sq. units) of the region

A=\{(x,y)\in R\times R \mid 0\le x\le 3, 0\le y\le 4, y\le x^{2}+3x\}

is:

(A) $\dfrac{26}{3}$
(B) $\dfrac{59}{6}$
(C) $\dfrac{53}{6}$
(D)8

Q139·MathematicsSingle correctJEE Main 2019

The area (in sq. units) of the region bounded by the parabola,

y = x^{2} + 2

and the lines,

y = x + 1

,

x = 0

and

x = 3

, is:

(A) $\dfrac{15}{4}$
(B) $\dfrac{21}{2}$
(C) $\dfrac{17}{4}$
(D) $\dfrac{15}{2}$

Q140·MathematicsSingle correctJEE Main 2019

The area (in sq. units) in the first quadrant bounded by the parabola,

y=x^{2}+1

, the tangent to it at the point (2, 5) and the coordinate axes is:

(A) $\frac{8}{3}$
(B) $\frac{37}{24}$
(C) $\frac{187}{24}$
(D) $\frac{14}{3}$

Q141·MathematicsSingle correctJEE Main 2019

The area (in sq. units) of the region bounded by the curve

x^{2}=4y

and the straight line

x=4y-2

is:

(A) $5/4$
(B) $9/8$
(C) $7/8$
(D) $3/4$

Q142·MathematicsSingle correctJEE Main 2019

If the area enclosed between the curves

y = kx^2

and

x = ky^2

, (

k > 0

), is 1 square unit. Then k is:

(A) $\dfrac{\sqrt{3}}{2}$
(B) $\dfrac{1}{\sqrt{3}}$
(C) $\sqrt{3}$
(D) $\dfrac{2}{\sqrt{3}}$

Q143·MathematicsSingle correctJEE Main 2019

The area (in sq. units) bounded by the parabola

y=x^{2}-1

, the tangent at the point (2, 3) to it and the y – axis is:

(A) $\dfrac{8}{3}$
(B) $\dfrac{32}{3}$
(C) $\dfrac{53}{3}$
(D) $\dfrac{14}{3}$

Q144·MathematicsSingle correctJEE Main 2019

The area of the region

A\{(x,y):0\leq y\leq x|x|+1

and

-1\leq x\leq 1\}

in sq. units, is:

(A) $\frac{2}{3}$
(B) $\frac{1}{3}$
(C)2
(D) $\frac{4}{3}$

Q145·MathematicsNumericalJEE Advanced 2018

A farmer

F_{1}

has a land in the shape of a triangle with vertices at

P(0, 0)

,

Q(1, 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

y = x^{n}

(n > 1)

. If the area of the region taken away by the farmer

F_{2}

is exactly 30% of the area of

\Delta

PQR, then the value of n is ______ .

Q146·MathematicsMultiple correctJEE Advanced 2017

If the line

x = \alpha

divides the area of region

R = \{(x, y) \in \mathbb{R}^{2} : x^{3} \le y \le x,\ 0 \le x \le 1\}

into two equal parts, then

(A) $0 < \alpha \le \dfrac{1}{2}$
(B) $\dfrac{1}{2} < \alpha < 1$
(C) $2\alpha^{4} - 4\alpha^{2} + 1 = 0$
(D) $\alpha^{4} + 4\alpha^{2} - 1 = 0$

Q147·MathematicsSingle correctJEE Advanced 2016

Area of the region

\left\{ (x, y) \in \mathbb{R}^{2} : y \geq \sqrt{|x + 3|}, 5y \leq x + 9 \leq 15 \right\}

is equal to

(A) $\frac{1}{6}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{5}{3}$

Q148·MathematicsIntegerJEE Advanced 2014

For a point

P

in the plane, let

d_1(P)

and

d_2(P)

be the distances of the point

P

from the lines

x - y = 0

and

x + y = 0

respectively. The area of the region

R

consisting of all points

P

lying in the first quadrant of the plane and satisfying

2 \le d_1(P) + d_2(P) \le 4

, is __________

Q149·MathematicsSingle correctJEE Advanced 2013

The area enclosed by the curves

y=\sin x+\cos x

and

y=|\cos x-\sin x|

over the interval

[0,\frac{\pi}{2}]

is

(A) $4(\sqrt{2}-1)$
(B) $2\sqrt{2}(\sqrt{2}-1)$
(C) $2(\sqrt{2}+1)$
(D) $2\sqrt{2}(\sqrt{2}+1)$

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