Question

What is the volume of the solid obtained by rotating the region bounded by the curve $x=y$ and $x=y^2$ about the y-axis?

• A. $\frac{\pi }{4}$
• B. $\frac{2\pi }{5}$
• C. $\frac{2\pi }{15}$
• D. $\frac{\pi }{15}$

Answer 1

The correct option is C $\frac{2\pi }{15}$

With Respect to $y$ axis

Outer radius $y=x$

Inner radius $y^2=x$

Volume $={\int }_0^1\pi [x_0^2-x_1^2]dx$

$={\int }_0^1\pi [y^2-y^4]dy$

$=\pi {[\frac{y^3}{3}-\frac{y^5}{5}]}_0^1$

$\frac{\pi }{15}[5-3]$

$=\frac{2\pi }{15}$