The volume of solid generated by revolving about the $y$ -axis the figure bounded by the parabola $y=x^{2}$ and $x=y^{2}$ is

• A

  $\frac{21}{5} \pi$
• B

  $\frac{24}{5} \pi$
• Correct answer
  C

  $\frac{2}{15} \pi$
• D

  $\frac{5}{24} \pi$

Solution

$V=\int_{0}^{1} \pi x^{2} d y$
$=\left|\pi \int_{0}^{1}\left(y^{4}-y\right) d y\right|$ $=\left|\pi\left[\frac{y^{5}}{5}-\frac{y^{2}}{2}\right]_{0}^{1}\right|$ $=\left|\pi\left[\frac{1}{5}-\frac{1}{3}\right]\right|$ $=\left|\frac{-2 \pi}{15}\right|=\frac{2 \pi}{15}$

Hence, the correct answer is option: 'C'.