Area Under Curves — JEE Main PYQs

Three points $\mathrm{O}(0,0),\mathrm{P}\left(\mathrm{a},{\mathrm{a}}^2\right),\mathrm{Q}\left(-\mathrm{b},{\mathrm{b}}^2\right),\mathrm{a}>0,\mathrm{b}>0$, are on the parabola $\mathrm{y}={\mathrm{x}}^2$. Let ${\mathrm{S}}_1$ be the area of the region bounded by the line …

The sum of squares of all possible values of k , for which area of the region bounded by the parabolas $2y^2=kx$ and $k{y}^2=2(y-x)$ is maximum, is equal to :

The area of the region
$\left\{(\mathrm{x},\mathrm{y}):{\mathrm{y}}^2\le 4\mathrm{x},\mathrm{x}<4,\frac{\mathrm{xy}(\mathrm{x}-1)(\mathrm{x}-2)}{(\mathrm{x}-3)(\mathrm{x}-4)}>0,\mathrm{x}\ne 3\right\}$
is

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to :

The area (in sq. units) of the part of circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha }{2\beta }-\frac{65}{2}+\frac{\alpha }{\beta }{\sin }^{-1}\left(\frac{12}{13}\right)$ where $\alpha ,\beta$ are coprime numbers. Then $\alpha +\beta$ is equal to

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 ______.

The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

The area of the region $\left\{(\mathrm{x},\mathrm{y}):{\mathrm{x}}^2\le \mathrm{y}\le 8-{\mathrm{x}}^2,\mathrm{y}\le 7\right\}$ is

If the area of the region
$S=\left\{(x,y):2y-y^2\le x^2\le 2y,x\ge y\right\}$ is equal to
$\frac{n+2}{n+1}-\frac{\pi }{n-1}$, then the natural number n is equal to

Let the area enclosed by the lines $x+y=2$, $y=0$, $x=0$ and the curve $f(x)=\min \left\{x^2+\frac{3}{4},1+[x]\right\}$ where $[x]$ denotes the greatest integer $\le x$, be A. Then the value of 12A is ______.

If the area of the region $\left\{(x,y):\left|x^2-2\right|\le y\le x\right\}$ is A, then $6\mathrm{A}+16\sqrt{2}$ is equal to ______.

Let $\mathrm{y}=\mathrm{p}(\mathrm{x})$ be the parabola passing through the points $(-1,0),(0,1)$ and $(1,0)$. If the area of the region $\left\{(x,y):(x+1{)}^2+(y-1{)}^2\le 1,y\le p(x)\right\}$ is A , then $12(\pi -4\mathrm{A})$ is equal to ____.

The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is

The area of the region $\left\{(x,y):x^2\le y\le \left|x^2-4\right|,y\ge 1\right\}$ is

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

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 $4(\pi +4\mathrm{A})$ is equal to ____

If A is the area in the first quadrant enclosed by the curve $\mathrm{C}:2{\mathrm{x}}^2-\mathrm{y}+1=0$, the tangent to C at the point (1,3) and the line $\mathrm{x}+\mathrm{y}=1$, then the value of 60A is

Area of the region $\left\{(x,y):x^2+(y-2{)}^2\le 4,x^2\ge 2y\right\}$ is

The value of $12{\int }_0^3\left|{\mathrm{x}}^2-3\mathrm{x}+2\right|\mathrm{dx}$ is

The area of the region given by $\left\{(\mathrm{x},\mathrm{y}):\mathrm{xy}\le 8,1\le \mathrm{y}\le {\mathrm{x}}^2\right\}$ is :

Let $\Delta$ be the area of the region
$\left\{(\mathrm{x},\mathrm{y})\in {\mathbb{R}}^2:{\mathrm{x}}^2+{\mathrm{y}}^2\le 21,{\mathrm{y}}^2\le 4\mathrm{x},\mathrm{x}\ge 1\right\}.$ Then $\frac{1}{2}\left(\Delta -21{\sin }^{-1}\frac{2}{\sqrt{7}}\right)$ is …

Let T and C respectively be the transverse and conjugate axes of the hyperbola $16x^2-y^2+64x+4y+44=0$. Then the area of the region above the parabola $x^2=y+4$, below the transverse axis T and on the right of the conjugate axis C is:

If ${\int }_{\frac{1}{3}}^3|{\log }_{e}x|dx=\frac{m}{n}{\log }_e\left(\frac{n^2}{e}\right)$, where $m$ and $n$ are coprime natural numbers, then ${\mathrm{m}}^2+{\mathrm{n}}^2-5$ is equal to ______.

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

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation ${\mathrm{x}}^2-\mathrm{px}+\frac{5}{4}\mathrm{p}=0$ are rational. Then the area of the region …

Let A be the area of the region

Then 540 A is equal to ____.

Let for $\mathrm{x}\in \mathrm{R}$


Then area bounded by the curve $\mathrm{y}=(\mathrm{fog})(\mathrm{x})$ and the lines $\mathrm{y}=0,2\mathrm{y}-\mathrm{x}=15$ is equal to

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 ____.

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

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

The area of the region given by
$\mathrm{A}=\left\{(\mathrm{x},\mathrm{y}):{\mathrm{x}}^2\le \mathrm{y}\le \min \{\mathrm{x}+2,4-3\mathrm{x}\}\right\}$ is :

The area bounded by the curves $y=\left|x^2-1\right|$ and $y=1$ is

The area of the region enclosed by
$\mathrm{y}\le 4{\mathrm{x}}^2,{\mathrm{x}}^2\le 9\mathrm{y}$ and $\mathrm{y}\le 4$, is equal to :

The odd natural number a, such that the area of the region bounded by $\mathrm{y}=1,\mathrm{y}=3,\mathrm{x}=0,\mathrm{x}={\mathrm{y}}^{\mathrm{a}}$ is $\frac{364}{3}$, equal to :

The area of the bounded region enclosed by the curve $\mathrm{y}=3-\left|\mathrm{x}-\frac{1}{2}\right|-|\mathrm{x}+1|$ and the x -axis is

If the area of the region $\left\{(\mathrm{x},\mathrm{y}):{\mathrm{x}}^{\frac{2}{3}}+{\mathrm{y}}^{\frac{2}{3}}\le 1,\mathrm{x}+\mathrm{y}\ge 0,\mathrm{y}\ge 0\right\}$ is A, then $\frac{256\mathrm{A}}{\pi }$ is

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

For real numbers $a,\mathrm{b}(a>\mathrm{b}>0)$, let
$\mathrm{Area}\{(x,y):x^2+y^2\le a^2$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge 1\}=30\pi$
and
Area $\{(x,y):x^2+y^2\ge b^2$ and …

The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to :-

The area enclosed by ${\mathrm{y}}^2=8\mathrm{x}$ 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 :

Let $f(x)=\max \{|x+1|,|x+2|,\dots ,|x+5|\}$. Then ${\int }_{-6}^{0}f(x)dx$ is equal to ______.

Let $A_1=\{(x,y):|x|\le y^2,|x|+2y\le 8\}$ and $A_2=\{(x,y):|x|+|y|\le k\}$. If $27$(Area $A_1$)$=5$(Area $A_2$), then k is equal to:

The area of the region $\left\{(x,y):|x-1|\le y\le \sqrt{5-x^2}\right\}$ is equal to :

Let S be the region bounded by the curves $\mathrm{y}={\mathrm{x}}^3$ and $y^2=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_1$ and $R_2$.

If max $\left\{R_1,R_2\right\}=R_2$, then $\frac{R_2}{R_1}$ is equal to ______.

The area (in sq. units) of the region enclosed between the parabola $y^2=2x$ and the line $x+y=4$ is ____.

The area of the region enclosed between the parabolas ${\mathrm{y}}^2=2\mathrm{x}-1$ and ${\mathrm{y}}^2=4\mathrm{x}-3$ is

The area of the region $S=\left\{(x,y):3x^2\le 4y\le 6x+24\right\}$ is ______.

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:

Let $T$ be the tangent to the ellipse $E:x^2+4y^2=5$ 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 $\alpha \sqrt{5}+\beta +\gamma {\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$, then …

The area of the region : $R=\left\{(x,y):5x^2\le y\le 2x^2+9\right\}$ is: