Application of Definite Integral to Areas

The area of the region between the curves  $y=\sqrt{\frac{1+\sin x}{\cos x}}$  and  $y=\sqrt{\frac{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-\sin x}{\cos x}}$  bounded by the lines  $x=0$  and  $x=\frac{\pi }{4}$  is:

If the area(in square units) of the region bounded between the line $x=2$ and the parabola $y^2=8x$ is $\frac{k}{3}$, then $k=$

The area (in sq. units) bounded by $y={\sin }^{2}x, x=\frac{\pi }{2}$ and $x=\pi$ is

The area bounded by $y={\sin }^{2}x, x=\frac{\pi }{2}$ and $x=\pi$ is

The area enclosed by ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ above $x$–axis in square units is

If the area bounded by circle $x^2+y^2=4$, $y$-axis and $x=1$ is $\sqrt{a}+\frac{2\pi }{b}$, find the value of $ab$.

Find the area of the region bounded by the curve $y=x^2$ and the line $y=4$.

If the area of the region bounded by the triangle whose vertices are $(1,0),(2,2)$ and $(3,1)$ is $k$ sq. units, then the value of $k$ is

The area of the region bounded by $y^2=24x$ and line $x=1$ is,

Find the area of the region bounded by $x^2=4y, y=2, y=4$ and the $y-$axis in the first quadrant.

 Find the area of the smaller region bounded by the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and the line $\frac{x}{3}+\frac{y}{2}=1$ by using the method of integration.

The area of the region in the first quadrant enclosed by the $x$-axis, the line $y=x$ and the circle $x^2+y^2=32$ is

Find the area bounded by the curves $y^2=4a^2(x-1)$ and the lines $x=1$,$y=4a$.

If the area of the region bounded by the curve $y=\sin x$, between $x=0$ and $x=2\pi$ is $k$ sq. units, then the value of $k$ is

If the area of the region bounded by curves $y=2x+1, y=3x+1$ and $x=4$ (in sq. units) is $k$, then the value of $k$ is

The maximum possible area bounded by the parabola $y=x^2+x+10$ and a chord of the parabola of length $1$ is

The area of the region bounded by the lines $x=1, x=2$, and the curves $x\left(y-e^x\right)=\sin x$ and $2xy=2\sin x+x^3$ is

The area bounded by the parabola $x^2=4y$ and line $y=3$ is equal to

The area of the region bounded by the curve $y^2=4x$ and the line $x=3$ is