Area between curves

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

Let $A_1=\left\{(x,y):|x|\le y^2,\ |x|+2y\le8\right\}$ and $A_2=\left\{(x,y):|x|+|y|\le k\right\}$. If $27\,(\text{Area }A_1)=5\,(\text{Area }A_2)$, then $k$ is equal to :

If the area of the region $\left\{(x,y):x^{2/3}+y^{2/3}\le1,\ x+y\ge0,\ y\ge0\right\}$ is A, then $\dfrac{256A}\pi$ is equal to ______.

The area of the region $S=\left\{(x,y):y^2\le8x,\ y\ge\sqrt2x,\ x\ge1\right\}$ is

The area enclosed by $y^2=8x$ and $y=\sqrt2x$ that lies outside the triangle formed by $y=\sqrt2x$, $x=1$, $y=2\sqrt2$, is equal to :

If $\displaystyle\int_0^2\left(\sqrt{2x}-\sqrt{2x-x^2}\right)dx=\int_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}2\right)dy+\int_1^2\left(2-\frac{y^2}2\right)dy+I$ then $I$ equals

For real numbers $a,b$ $(a>b>0)$, let
$$\text{Area}\left\{(x,y):x^2+y^2\le a^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge1\right\}=30\pi$$
and
$$\text{Area}\left\{(x,y):x^2+y^2\ge b^2\text{ and }\frac{x^2}{a^2}+\frac{y^2}{b^2}\le1\right\}=18\pi$$
Then the value of $(a-b)^2$ is equal to ______.

If for some $\alpha>0$, the area of the region $\{(x,y):|x+\alpha|\le y\le2-|x|\}$ is equal to $\dfrac32$, then the area of the region $\{(x,y):0\le y\le x+2\alpha,\ |x|\le1\}$ is equal to ______.

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

Let the locus of the centre $(\alpha,\beta)$, $\beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also touches the $x$-axis be L. Then the area bounded by L and the line $y=4$ is :

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

The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to

The area of the region enclosed by $y\le4x^2$, $x^2\le9y$ and $y\le4$, is equal to :

The area enclosed by the curves $y=\log_e\left(x+e^2\right)$, $x=\log_e\left(\dfrac2y\right)$ and $x=\log_e2$, above the line $y=1$ is :

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

If the area enclosed by the curves $y=x^2$, $y^3=x$, $x=-1$ and $y=x^2$, $y^3=x$, $x=1$, above the line $x=2y$ is $\dfrac n{n+1}$, then n is equal to _______.

The area of the region
$$\left\{(x, y) : 0 \le x \le \frac94,\quad 0 \le y \le 1,\quad x \ge 3y,\quad x + y \ge 2\right\}$$
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 the functions $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be defined by

$$f(x) = e^{x-1} - e^{-|x-1|}\quad\text{and}\quad g(x) = \frac12\left(e^{x-1} + e^{1-x}\right).$$

Then the area of the region in the first quadrant bounded by the curves $y = f(x)$, $y = g(x)$ and $x = 0$ is

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