Area Under Curves — JEE Main PYQs

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:

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

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 ____

The area bounded by the lines $y=\Vert x-1|-2|$ is ____

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,

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 $12m$ is equal to ____.

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 $(\alpha +\beta -2\gamma {)}^2$ is equal to …

The area (in sq. units) of the region, given by the set
$\left\{(x,y)\in R\times R\mid x\ge 0,2x^2\le y\le 4-2x\right\}$ is :

The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :

The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0,x=\frac{\pi }{2}$, is:

The area bounded by the curve $4y^2=x^2(4-x)(x-2)$ is equal to :

Let $f:[-3,1]\to R$ be given as
$f(x)=\{\begin{array}{llc}\min \{(x+6),& x^2\},& -3\le x<0\\ \max \{\sqrt{x},& x^2\},& 0\le x\le 1\end{array}$
If the area bounded by $y=f(x)$ and $x$-axis is $A$, then the value of $6A$ is equal to ____.

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 :

The area (in sq. units of the region $\left\{(x,y)\in R^2\mid 4x^2\le y\le 8x+12\right\}$ is:

For $\mathrm{a}>0$, let the curves ${\mathrm{C}}_1:{\mathrm{y}}^2=\mathrm{ax}$ and ${\mathrm{C}}_2:{\mathrm{x}}^2=\mathrm{ay}$ intersect at origin O and a point P . Let the line $\mathrm{x}=\mathrm{b}(0<\mathrm{b}<\mathrm{c})$ intersect the chord OP and the x - axis at points Q …

The area (in sq. units) of the region $\left\{(x,y)\in R^2\le y\le 3-2x\right\}$, is

Given : $f(x)=\{\begin{array}{cc}x,& 0\le x<\frac{1}{2}\\ \frac{1}{2},& x=\frac{1}{2}\\ 1-x,& \frac{1}{2}<x\le 1\end{array}$ and $g(x)={\left(x-\frac{1}{2}\right)}^2,x\in R$. Then the area (in sq. units) of the region bounded by the curves, $y=f(x)$ a …

The area (in sq. units) of the region $\mathrm{A}=\{(\mathrm{x},\mathrm{y}):(\mathrm{x}-1)[\mathrm{x}]\le \mathrm{y}\le 2\sqrt{\mathrm{x}},0\le \mathrm{x}\le 2\}$, where [t] denotes the greatest integer function, is:

The area (in sq. units) of the region
$\left\{(x,y):0\le y\le x^2+1,0\le y\le x+1,\frac{1}{2}\le x\le 2\right\}$ is:

The area (in sq. units) of the region enclosed by the curves $y=x^2-1$ and $y=1-x^2$ is equal to

The area (in sq. units) of the region $\mathrm{A}=\left\{(\mathrm{x},\mathrm{y}):|\mathrm{x}|+|\mathrm{y}|\le 1,2{\mathrm{y}}^2\ge |\mathrm{x}|\right\}$ is:

The area (in sq. units) of the region $\mathrm{A}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{R}\times \mathrm{R}\mid 0\le \mathrm{x}\le 3,0\le \mathrm{y}\le 4,\mathrm{y}\le {\mathrm{x}}^2+3\mathrm{x}\right\}$ is:

Let $S(\alpha )=\left\{(x,y):y^2\le x,0\le x\le \alpha \right\}$ and $A(\alpha )$ is area of the region $S(\alpha )$. If for a $\lambda ,0<\lambda <4,\mathrm{A}(\lambda ):\mathrm{A}(4)=2:5$ then $\lambda$ equals:

The area (in sq. units) of the region $A=\left\{(x,y):x^2\le y\le x+2\right\}$ is:

If the area enclosed between the curves $y=k{x}^2$ and $x=k{y}^2,(k>0)$, is 1 square unit. Then k is:

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:

If the area (in sq. units) of the region $\left\{(x,y):y^2\le 4x,x+y\le 1,x\ge 0,y\ge 0\right\}$ is $a\sqrt{2}+b$, then $\mathrm{a}-\mathrm{b}$ is equal to:

The area(in sq. units) of the region bounded by the curves $y=2^x$ and $y=|x+1|$ in the first quadrant is

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

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:

If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x$, $\lambda >0$, is $\frac{1}{9}$, then $\lambda$ is equal to :

The area (in sq. units) of the region $\mathrm{A}=\left\{(\mathrm{x},\mathrm{y}):\frac{{\mathrm{y}}^2}{2}\le x\le y+4\right\}$ is:

The area of the region $\mathrm{A}\{(\mathrm{x},\mathrm{y}):0\le \mathrm{y}\le \mathrm{x}|\mathrm{x}|+1$ and $-1\le \mathrm{x}\le 1\}$ in sq. units, is:

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:

Let $\mathrm{g}(x)=\cos x^2,f(x)=\sqrt{x}$, and $\alpha ,\beta (\alpha <\beta )$ be the roots of the quadratic equation $18x^2-9\pi x+{\pi }^2=0$. Then the area (in sq. units) bounded by the curve $y=\left(g_{\circ }f\right)(x)$ and the lines $x=\alpha ,x=\beta$ and …

The area (in sq. units) of the region $\{(x,y):x⩾0,x+y\le 3,x^2\le 4y$ and $y\le 1+\sqrt{x}\}$ is :

The area (in sq. units) of the region $\{(x,y):y^2\ge 2x$ and $x^2+y^2\le 4x,x\ge 0,y\ge 0\}$ is:

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

The area of the region described by $\mathrm{A}=\{(x,y):x^2+y^2\le 1$ and $y^2\le 1-x\}$ is

The area (in square units) bounded by the curves $y=\sqrt{x}$, $2y-x+3=0$, $x$-axis, and lying in the first quadrant is

The area bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9y$, and the straight line $y=2$ is

The area of the region enclosed by the curves $y=x,x=e,y=\frac{1}{x}$ and the positive $x$-axis is

The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3\pi }{2}$ is

The area of the region bounded by the parabola $(y-2{)}^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the $x$-axis is

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

The area enclosed between the curves $y^2=x$ and $y=|x|$ is

The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is

The parabolas $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes. If $S_1,S_2,S_3$ are respectively the areas of these parts numbered from top to bottom; then …

The area of the region bounded by the curves $y=|x-2|,x=1,x=3$ and the $x$-axis is