Area Under the Curve Questions - JEE

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:

The value of 'a' (a > 0) for which the area bounded by the curve $y = \frac{x-1}{6-x^2}$, $y = 0$, $x = a$ and $x = 2a$ has the least value, is

The region represented by |x − y| ≤ 2 and |x + y| ≤ 2 is bounded by a

The area (in sq. units) bounded by $y^2=x$ and $x^2=y$. [JEE Main 2019]

Given the region $\dfrac{y^2}{2} \le x \le y + 4$, find the area of the region (in square units).

Area of the region $\{(x,y)\,:\,y\ge0,\,y\le x,\,y\le 2-x\}$. [JEE Main 2023]

Area of $\{(x,y)\,:\,x^2\le y\le\sqrt{x}\}$. [JEE 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 ___.

Find the area enclosed between the curves $ x + 2y^2 = 0 $ and $ x + 3y^2 = 1 $.

The area enclosed by $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over $[0,\pi/2]$ is: [MAU009]

The region bounded by the curves $y = 2^x$ and $y = |x + 1|$ has area equal to:

If the area enclosed between the curves y = kx² and x = ky², (k > 0), is 1 square unit. Then k is:

Given the region bounded by the curves $y = x^2$, $y = \dfrac{1}{x}$ and the lines $y = 0$ and $x = t$ $(t > 1)$. If the area bounded by these curves is 1, then $t$ equals:

Sketch the region bounded by the curves $y = \log_e x$ and $y = (\log_e x)^2$. Also find the area of the region.

The area bounded by $y = x^2 + 2$ and $y = 2|x| - \cos x$ is of the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p - q$.

The area of the region $\{(x,y)\,:\,x^2+y^2\le1\le x+y\}$. [JEE Main 2020]

The area enclosed by the closed curve C given by the differential equation $\dfrac{dy}{dx}+\dfrac{x+a}{y-2}=0$, $y(1)=0$ is $4\pi$. Let P and Q be the points of intersection of C and the y-axis. If normals at P and Q on C intersect x-axis at R and S respectively, then the length of RS is:

The area of the region consisting of all points $(x, y)$ so that $x^2 + y^2 \leq 1$ and $|x| + |y| \leq 1$ is

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 ___.

Let $A=\{(x,y)\in\mathbb{R}^2:y\geq0,\,2x\leq y\leq\sqrt{4-(x-1)^2}\}$ 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:

The area of the region $A=\left\{(x,y):|\cos x-\sin x|\leq y\leq\sin x,\,0\leq x\leq\dfrac{\pi}{2}\right\}$ is:

21. Let $A_k$ be the finite area bounded by the line $y = kx + k$ and the parabola $y = x^2$, where $k$ is a positive real number. The value of $\displaystyle\lim_{k \to \infty} \dfrac{A_k}{k^3}$ equals:

Let $A_1$ be the area of the region bounded by the curves $y = \sin x, y = \cos x$ 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, (JEE Main 2021)

The area (in sq. units) bounded between the parabola $y=x^2$ and the line $y=x$ is: [MAU007]

If the area enclosed between $f(x) = \min\left\{\cos^{-1}(\cos x), \cot^{-1}(\cot x)\right\}$ and the x-axis in $x \in \left(\frac{k\pi}{2}, \frac{(k+1)\pi}{2}\right)$ where $k \in \mathbb{N}$, then k is equal to

Let A be the area of the region $\{(x,y):y\geq x^2,\,y\geq(1-x)^2,\,y\leq2x(1-x)\}$. Then $540A$ is equal to ___.

Let the area of the region $\{(x,y):|2x-1|\leq y\leq|x^2-x|,\,0\leq x\leq1\}$ be $A$. Then $(6A+11)^2$ is equal to ___.

Let $A = \{(x, y) : y^2 \leq 4x,\ y - 2x \geq -4\}$. The area (in square units) of the region $A$ is

Let the line $x=-1$ divide the area of the region $\{(x,y):1+x^2\leq y\leq3-x\}$ in the ratio $m:n$, $\gcd(m,n)=1$. Then $m+n$ is equal to

Let $S(\alpha) = \{(x, y) : y^2 \leq x,\ 0 \leq x \leq \alpha\}$ 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:

Consider a square with vertices at (1,1), (1,-1), (-1,-1) and (-1,1). Let S be the region consisting of all those points inside the square which are nearer to the origin than any side. Sketch the region S and find its area.

If ABC is an isosceles triangle inscribed in a circle of radius r. If AB = AC and h is altitude from A to BC then the triangle ABC has perimeter $P = 2(\sqrt{2hr - h^2} + \sqrt{2hr})$, calculate area A and $\lim_{h \to 0} \dfrac{A}{P^3}$.

The area between the parabola $x=4y-y^2$ and the line $x=y$ is: [MAU014]

The area of the region enclosed by $y^2\le4x$ and $x\le4$. [JEE Main 2022]

Let g(x) = cos x², f(x) = √x and α, β (α < β) be the roots of the quadratic equation 18x² − 9πx + π² = 0. Then, the area (in sq. units) bounded by the curve y = (g ∘ f)(x) and the lines x = α, x = β and y = 0 is

The area (in sq. units) of the region bounded by the curve x² = 4y and the straight line x = 4y − 2 is:

The area of the region of the xy plane defined by the inequality $|x| + |y| + |x + y| \leq 1$ is

The area bounded by the curve y = f(x), the coordinate axes and the line x = x₁ is given by x₁eˣ¹ - 1. Therefore f(x) equals:

Find the area of the region lying inside $x^2 + (y-1)^2 = 1$ and outside $c^2x^2 + y^2 = c^2$, where $c = \sqrt{2} - 1$.

The area bounded by y = x² + 2 and y = 2|x| – cos x is equal to

The area enclosed by the curve $x = a\cos^3 t$, $y = b\sin^3 t$, is

If area bounded by the line y = x, curve y = f(x) and lines x = 1, x = t, is t/2(2t – 1 + 1/t + t) then f(x) =

Suppose y = f(x) and y = g(x) are two functions whose graphs intersect at the three points (0, 4), (2, 2) and (4, 0) with f(x) > g(x) for 0 and f(x) for 2 . If ∫₀⁴ [f(x) - g(x)]dx = 10 and ∫₂⁴ [g(x) - f(x)]dx = 5, the area between two curves for 0 is:

The area enclosed between the curves $|x| + |y| \geq 2$ and $y^2 = 4\left(1 - \frac{x^2}{9}\right)$ is:

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 ___.

First draw the graph of $[x] + [y] = 3$, where $[\cdot]$ denotes the greatest integer function. Then find the area of the graph of $\lfloor |x| \rfloor + \lfloor |y| \rfloor = 3$.

The area of the region above the $x$-axis bounded by the curve $y = \tan x,\ 0 \leq x \leq \dfrac{\pi}{2}$ and the tangent to the curve at $x = \dfrac{\pi}{4}$ is

Area of $y=|x^2-4|$ from $x=-3$ to $x=3$. [JEE Main 2022]

Find the area of the region bounded by the square ABCD with area 2 sq units and a circle with radius $\frac{1}{2}$ sq units inscribed in it.

Area enclosed by the curve $|x + y - 1| + |2x + y + 1| = 1$ is

Area Under the Curve Questions (274)