Area under Curves

19 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q2·MCQ·Easy–moderate
Open
Area between curves
The area of the region, inside the ellipse $x^2 + 4y^2 = 4$ and outside the region bounded by the curves $y = \lvert x\rvert - 1$ and $y = 1 - \lvert x\rvert$ , is:
1. (1) $2(\pi - 1)$
2. (2) $2\pi - \dfrac12$
3. (3) $3(\pi - 1)$
4. (4) $2\pi - 1$
MathsJEE Main 21 Jan 2026, Shift 2·Q11·MCQ·Easy–moderate
Open
Area between curves
If the area of the region $\{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0, y \ge 0\}$ is $\dfrac{\alpha}{\beta}$ , $\alpha, \beta, \in \mathbb{N}$ , $\gcd(\alpha, \beta) = 1$ , then the value of $(\alpha + \beta)$ is:
1. (1)73
2. (2)85
3. (3)91
4. (4)67
MathsJEE Main 22 Jan 2026, Shift 1·Q3·MCQ·Easy–moderate
Open
Area between curves
Let the line $x = -1$ divide the area of the region $\{(x, y) : 1 + x^2 \le y \le 3 - x\}$ in the ratio $m : n$ , $\gcd(m, n) = 1$ . Then $m + n$ is equal to
1. (1)25
2. (2)28
3. (3)26
4. (4)27
MathsJEE Main 22 Jan 2026, Shift 2·Q12·MCQ·Moderate
Open
Area between curves
The area of the region $A = \{(x, y) : 4x^2 + y^2 \le 8 \text{ and } y^2 \le 4x\}$ is:
1. (1) $\dfrac{\pi}{2} + 2$
2. (2) $\pi + \dfrac23$
3. (3) $\pi + 4$
4. (4) $\dfrac{\pi}{2} + \dfrac13$
MathsJEE Main 23 Jan 2026, Shift 1·Q22·Numerical·Moderate
Open
Area bounded by curves
Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$ , lines $x = 0$ , $x = \dfrac{3\pi}{2}$ , and the $x$ -axis be $A$ . Then, $A + A^2$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 2·Q12·MCQ·Moderate
Open
Area between two circles
The area of the region enclosed between the circles $x^2 + y^2 = 4$ and $x^2 + (y - 2)^2 = 4$ is:
1. (1) $\dfrac23(2\pi - 3\sqrt3)$
2. (2) $\dfrac43(2\pi - 3\sqrt3)$
3. (3) $\dfrac43(2\pi - \sqrt3)$
4. (4) $\dfrac23(4\pi - 3\sqrt3)$
MathsJEE Main 24 Jan 2026, Shift 1·Q19·MCQ·Moderate
Open
Area between curves
Let $A_1$ 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 $A_2$ 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 $A_1 - A_2$ is equal to
1. (1) $\dfrac23\left(2\sqrt2 + 1\right)$
2. (2) $\dfrac23\left(4\sqrt2 + 1\right)$
3. (3) $\dfrac23\left(\sqrt2 + 1\right)$
4. (4) $\dfrac23\left(3\sqrt2 + 1\right)$
MathsJEE Main 24 Jan 2026, Shift 2·Q12·MCQ·Moderate
Open
Area between curves
Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0$ , $x = 1$ , $y^2 = x$ and $y = \lvert\alpha x - 5\rvert - \lvert 1 - \alpha x\rvert + \alpha x^2$ . Then $(f(0) + f(1))$ is equal to
1. (1)9
2. (2)14
3. (3)7
4. (4)12
MathsJEE Main 28 Jan 2026, Shift 1·Q13·MCQ·Moderate
Open
Area bounded by curves
The area of the region $R = \{(x, y) : xy \le 8,\ 1 \le y \le x^2,\ x \ge 0\}$ is
1. (1) $\dfrac{1}{3}\left(49\log_e(2) - 15\right)$
2. (2) $\dfrac{2}{3}\left(20\log_e(2) + 9\right)$
3. (3) $\dfrac{2}{3}\left(24\log_e(2) - 7\right)$
4. (4) $\dfrac{1}{3}\left(40\log_e(2) + 27\right)$
MathsJEE Main 28 Jan 2026, Shift 2·Q11·MCQ·Moderate
Open
Area between parabolas and line
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 $y = \alpha x$ , $\alpha > 0$ and $P_1$ , then $\alpha$ is equal to :
1. (1)8
2. (2)15
3. (3)12
4. (4)6
MathsJEE Main 2 Apr 2026, Shift 2·Q24·Numerical·Moderate
Open
Area between hyperbola and line
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 .
MathsJEE Main 4 Apr 2026, Shift 1·Q18·MCQ·Moderate
Open
Area between curves
The area of the region $\{(x, y) : y \le \pi - \lvert x\rvert,\ y \le \lvert x\sin x\rvert,\ y \ge 0\}$ is:
1. (1) $1 + \dfrac{\pi^2}{8}$
2. (2) $2 + \dfrac{\pi^2}{4}$
3. (3) $\dfrac{\pi^2}{8} - 1$
4. (4) $4 + \dfrac{\pi^2}{2}$
MathsJEE Main 4 Apr 2026, Shift 2·Q18·MCQ·Easy–moderate
Open
Area between two parabolas
The area of the region bounded by the curves $x + 3y^2 = 0$ and $x + 4y^2 = 1$ is equal to:
1. (1) $\dfrac13$
2. (2) $\dfrac23$
3. (3) $\dfrac43$
4. (4) $\dfrac53$
MathsJEE Main 5 Apr 2026, Shift 1·Q16·MCQ·Moderate
Open
Area between curves
The area of the region $R = \{(x, y) : xy \le 27,\ 1 \le y \le x^2,\ x \ge 0\}$ is equal to:
1. (1) $78\log_e 3 - \dfrac{52}{3}$
2. (2) $54\log_e 3 - \dfrac{52}{3}$
3. (3) $54\log_e 3 - \dfrac{26}{3}$
4. (4) $54\log_e 3 + \dfrac{26}{3}$
We corrected a printing error in the original paper
Assumption “ $x \ge 0$ ” added to the region; the source leaves it unstated (for $x \le -1$ every point with $1 \le y \le x^2$ satisfies $xy \le 27$ , so the region as printed has infinite area).
MathsJEE Main 5 Apr 2026, Shift 2·Q24·Numerical·Tough
Open
Area between curves
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) + 3f\left(\dfrac{\pi}{2} - x\right) = \sin x$ , $x \in \mathbb{R}$ . Let maximum value of $f$ on $\mathbb{R}$ be $\alpha$ . If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3$ , $\beta > 0$ , is $\alpha^2$ , then $30\beta^3$ is equal to .
MathsJEE Main 6 Apr 2026, Shift 1·Q19·MCQ·Moderate
Open
Area bounded by parabola and line
The area of the region $\{(x, y) : 0 \le y \le 6 - x,\ y^2 \ge 4x - 3,\ x \ge 0\}$ is:
1. (1)8
2. (2)9
3. (3)12
4. (4)15
MathsJEE Main 6 Apr 2026, Shift 1·Q20·MCQ·Moderate
Open
Area between curves
Let $e$ be the base of natural logarithm and let $f : \{1, 2, 3, 4\} \to \{1, e, e^2, e^3\} \quad\text{and}\quad g : \{1, e, e^2, e^3\} \to \left\{1, \dfrac12, \dfrac13, \dfrac14\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac12\right)\right\}\right]^x,$ then the area of the region $R = \{(x, y) : x^2 \le y \le \phi(x),\ 0 \le x \le 1\}$ is:
1. (1) $\dfrac{3 - \log_e(2)}{3\log_e(2)}$
2. (2) $\dfrac{1}{3\log_e(2)}$
3. (3) $3 + \log_e(2)$
4. (4) $\dfrac{3 + \log_e(2)}{2 + \log_e(3)}$
MathsJEE Main 6 Apr 2026, Shift 2·Q19·MCQ·Easy–moderate
Open
Area between parabola and line
The area of the region $\{(x, y) : x^2 - 8x \le y \le -x\}$ is:
1. (1) $\dfrac{343}{6}$
2. (2) $\dfrac{637}{6}$
3. (3) $\dfrac{437}{6}$
4. (4) $\dfrac{523}{6}$
MathsJEE Main 8 Apr 2026, Shift 2·Q18·MCQ·Moderate
Open
Area between curves
Let $f : (1, \infty) \to \mathbb{R}$ be function defined as $f(x) = \dfrac{x-1}{x+1}$ . Let $f^{i+1}(x) = f\left(f^{i}(x)\right)$ , $i = 1, 2, \ldots, 25$ , 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:
1. (1) $\dfrac18 + \log_e 2$
2. (2) $\dfrac14 + \log_e 2$
3. (3) $\dfrac56 + 3\log_e 2$
4. (4) $\dfrac56 + \log_e 2$
We corrected a printing error in the original paper
Source printed “where $f^{1}(x) = (x)$ ” (the $f$ is missing).