Area Under Curves — JEE Main PYQs

The area of the region, inside the ellipse $x^2+4y^2=4$ and outside the region bounded by the curves $\mathrm{y}=|\mathrm{x}|-1$ and $\mathrm{y}=1-|\mathrm{x}|$, is :

If the area of the region $\{(x,y):1-2x\le y\le 4-x^2,x\ge 0,y\ge 0\}$ is $\frac{\alpha }{\beta }$, $\alpha ,\beta \in \mathbf{N}$, $\gcd (\alpha ,\beta )=1$, then the value of $(\alpha +\beta )$ is :

Let the line $\mathrm{x}=-1$ divide the area of the region $\left\{(x,y):1+x^2\le y\le 3-x\right\}$ in the ratio $m:n,\gcd (\mathrm{m},\mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to

The area of the region
$A=\{(x,y):4x^2+y^2\le 8\text{ and }{y}^2\le 4x\}$ is :

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 ______.

The area of the region enclosed between the circles $x^2+y^2=4$ and $x^2+(y-2{)}^2=4$ is :

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. Th …

Let $f(\alpha )$ denote the area of the region in the first quadrant bounded by $\mathrm{x}=0,\mathrm{x}=1,{\mathrm{y}}^2=\mathrm{x}$ and $\mathrm{y}=|\alpha x-5|-|1-\alpha x|+\alpha x^2$. Then $(f(0)+f(1))$ is equal to

The area of the region
$R=\left\{(x,y):xy\le 8,1\le y\le x^2,x\ge 0\right\}$ is

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 ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ is six times the area of the bounded region enclosed between the line $\mathrm{y}=\alpha \mathrm{x},\alpha >0$ and …

The area of the region bounded by the curves
$x+3y^2=0$ and $x+4y^2=1$ is equal to :

The area of the region $\{(x,y):y\le \pi -|x|,y\le |x\sin x|,y\ge 0\}$ is :

The area of the region
$\mathrm{R}=\left\{(\mathrm{x},\mathrm{y}):\mathrm{xy}\le 27,1\le \mathrm{y}\le {\mathrm{x}}^2\right\}$ is equal to :

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a function such that $\mathrm{f}(\mathrm{x})+3\mathrm{f}\left(\frac{\pi }{2}-\mathrm{x}\right)=\sin \mathrm{x},\mathrm{x}\in \mathrm{R}$. Let maximum value of $f$ on $R$ be $\alpha$. If the area of the region boun …

The area of the region $\left\{(x,y):x^2-8x\le y\le -x\right\}$ is:

The area of the region $\{(\mathrm{x},\mathrm{y}):0\le \mathrm{y}\le 6-\mathrm{x}$, ${\mathrm{y}}^2\ge 4\mathrm{x}-3,\mathrm{x}\ge 0\}$ is :

Let e be the base of natural logarithm and let $f:\{1,2,3,4\}\to \left\{1,\mathrm{e},{\mathrm{e}}^2,{\mathrm{e}}^3\right\}$ and $\mathrm{g}:\left\{1,\mathrm{e},{\mathrm{e}}^2,{\mathrm{e}}^3\right\}\to \left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\right\}$ …

Let $\mathrm{f}:(1,\infty )\to \mathbf{R}$ be function defined as $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}+1}$. Let ${\mathrm{f}}^{\mathrm{i}+1}(\mathrm{x})=\mathrm{f}\left({\mathrm{f}}^{\mathrm{i}}(\mathrm{x})\right)$,
$\mathrm{i}=1,2,\dots ,25$, whe …

If the area of the region bounded by $16x^2-9y^2=144$ and $8\mathrm{x}-3\mathrm{y}=24$ is A , then $3\left(\mathrm{A}+6{\log }_{\mathrm{e}}(3)\right)$ is equal to ______.

The area of the region, inside the circle $(x-2\sqrt{3}{)}^2+y^2=12$ and outside the parabola $y^2=2\sqrt{3}x$ is

Let the function, $f(x)=\{\begin{array}{ll}-3a{x}^2-2,& x<1\\ a^2+bx,& x\ge 1\end{array}$ be differentiable for all $\mathrm{x}\in \mathbf{R}$, where $\mathrm{a}>1,\mathrm{b}\in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $\mathrm{y}=-20$ is …

The area of the region enclosed by the curves $y=x^2-4x+4$ and $y^2=16-8x$ is :

If the area of the larger portion bounded between the curves $x^2+y^2=25$ and $y=|x-1|$ is $\frac{1}{4}(b\pi +c)$, $\mathrm{b},\mathrm{c}\in \mathbb{N}$, then $\mathrm{b}+\mathrm{c}$ is equal to

If the area of the region
$\left\{(\mathrm{x},\mathrm{y}):-1\le \mathrm{x}\le 1,0\le \mathrm{y}\le \mathrm{a}+{\mathrm{e}}^{|\mathrm{x}|}-{\mathrm{e}}^{-\mathrm{x}},\mathrm{a}>0\right\}$ is $\frac{e^2+8e+1}{e}$, then the value of $a$ is :

Let the area enclosed between the curves $|\mathrm{y}|=1-{\mathrm{x}}^2$ and ${\mathrm{x}}^2+{\mathrm{y}}^2=1$ be $\alpha$. If $9\alpha =\beta \pi +\gamma ;\beta ,\gamma$ are integers, then the value of $|\beta -\gamma |$ equals

Let the area of the region $\left\{(x,y):2y\le x^2+3,\mathrm{y}+|\mathrm{x}|\le 3,\mathrm{y}\ge |\mathrm{x}-1|\right\}$ be A. Then 6A is equal to:

The area of the region bounded by the curves $\mathrm{x}\left(1+{\mathrm{y}}^2\right)=1$ and ${\mathrm{y}}^2=2\mathrm{x}$ is :

The area of the region enclosed by the curves $y=e^x,y=\left|e^x-1\right|$ and $y$-axis is:

The area of the region $\left\{(\mathrm{x},\mathrm{y}):{\mathrm{x}}^2+4\mathrm{x}+2\le \mathrm{y}\le |\mathrm{x}+2|\right\}$ is equal to

The area of the region $\{(x,y):|x-y|\le y\le 4\sqrt{x}\}$ is

If the area of the region
$\left\{(\mathrm{x},\mathrm{y}):\left|4-{\mathrm{x}}^2\right|\le \mathrm{y}\le {\mathrm{x}}^2,\mathrm{y}\le 4,\mathrm{x}\ge 0\right\}$
is …

The area of the region bounded by the curve $y=\max \{|x|,x|x-2|\}$, then x-axis and the lines $x=-2$ and $x=4$ is equal to ______.

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:{\mathrm{y}}^2=\mathrm{x}-2$ at the point B in the first quadrant. The area, of the region bounded by the line AB, parabola P and the x-axis, is :-

If the area of the region $\{(x,y):|x-5|\le y\le 4\sqrt{x}\}$ is A , then 3 A is equal to

If the area of the region
$\left\{(\mathrm{x},\mathrm{y}):1+{\mathrm{x}}^2\le \mathrm{y}\le \min \{\mathrm{x}+7,11-3\mathrm{x}\}\right\}$ is A , then 3 A is equal to

If the area of the region bounded by the curves $y=4-\frac{x^2}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$, then $6\alpha$ equals

Let the area of the bounded region $\left\{(\mathrm{x},\mathrm{y}):0\le 9\mathrm{x}\le {\mathrm{y}}^2,\mathrm{y}\ge 3\mathrm{x}-6\right\}$ be A . Then 6 A is equal to ______

The area (in sq. units) of the region {(x, y) : 0 ≤ y ≤ 2|x| + 1, 0 ≤ y ≤ x² + 1, |x| ≤ 3} is:

One of the points of intersection of the curves $y=1+3x-2x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2},2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(\ell \sqrt{5}+m)-n{\log }_e(1+\sqrt{5})$, where $\ell ,m,n\in N$. Then …

The area (in sq. units) of the region described by $\{(x,y):y^2\le 2x\text{ and }y\ge 4x-1\}$ is

The area of the region enclosed by the parabolas $y=x^2-5x$ and $y=7x-x^2$ is ____.

Let the area of the region enclosed by the curve $\mathrm{y}=\min \{\sin \mathrm{x},\cos \mathrm{x}\}$ and the x -axis between $\mathrm{x}=-\pi$ to $\mathrm{x}=\pi$ be A . Then ${\mathrm{A}}^2$ is equal to ______ .

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 :

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 :

The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $\mathrm{y}=\mathrm{x}$ is :

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 ______.

The area of the region enclosed by the parabola $y=4x-x^2$ and $3y=(x-4{)}^2$ is equal to

The area (in square units) of the region bounded by the parabola ${\mathrm{y}}^2=4(\mathrm{x}-2)$ and the line $\mathrm{y}=2\mathrm{x}-8$

If the area of the region
$\left\{(\mathrm{x},\mathrm{y}):0\le \mathrm{y}\le \min \left\{2\mathrm{x},6\mathrm{x}-{\mathrm{x}}^2\right\}\right\}$ is A , then 12 A is equal to ______

Let the area of the region $\{(\mathrm{x},\mathrm{y}):\mathrm{x}-2\mathrm{y}+4\ge 0,x+2y^2\ge 0,x+4y^2\le 8,y\ge 0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $\mathrm{m}+\mathrm{n}$ is equal to ______.