Area Under Curves

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

The area bounded by the curve $y=\left|x^2-9\right|$ and the line $y=3$ is :

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 a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is

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 of the bounded region enclosed by the curve $y=3-\left|x-\dfrac12\right|-|x+1|$ and the $x$-axis 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 odd natural number a, such that the area of the region bounded by $y=1$, $y=3$, $x=0$, $x=y^a$ is $\dfrac{364}3$, is equal to :

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 :

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_1$ is twice the area $A_2$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point.

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 :