Solid of Revolution

Q.

The volume of the solid formed by rotating the area enclosed between the curve $y^2=4x$, $x=4$ and $x=5$ about the $x-$axis is (in cubic units)

1. $18\mathrm{\pi }$

2. $36\mathrm{\pi }$

3. $9\mathrm{\pi }$

4. $24\mathrm{\pi }$

Q. A parametric curve defined by $x=cos\left(\frac{\pi u}{2}\right),y=sin\left(\frac{\pi u}{2}\right)$ in the range $0\le u\le 1$ is rotated about the $X-$axis by $360$ degrees. Area of the surface generated is

1. $\frac{\pi }{2}$
2. $\pi$
3. $2\pi$
4. $4\pi$

Q. A curve y = $\frac{1}{2\sqrt{x}}$ is allowed to revolve around x axis. The volume of solid of revolution for $3\le x\le 4$ is

1. $x(\sqrt{3}+\frac{2}{\sqrt{3}})$
2. $\frac{\pi }{2}ln\left(\frac{8}{3}\right)$
3. $\frac{\pi }{4}ln\left(\frac{4}{3}\right)$
4. $\frac{\pi }{2}(\frac{1}{2\sqrt{2}}-\frac{1}{\sqrt{3}})$

Q. Consider the hemi-spherical tank of radius $13m$ as shown in the figure (not drawn to scale). What is the volume of water $\left(in{m}^3\right)$ when the depth of water at the centre of the tank is $6m?$

1. $78\pi$
2. $468\pi$
3. $156\pi$
4. $396\pi$

Q. The parabolic arc $y=\sqrt{x},1\le \times \le 2$ is resloved around the $x-$axis. The volume of the solid of revolution is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\frac{3\pi }{4}$
4. $\frac{3\pi }{2}$