Question

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 $
• C. $ \frac{2}{15} \pi $
• D. $ \frac{5}{24} \pi $

Answer 1

Answer: (C)

$V=\int \limits_{0}^{1} \pi x^{2} d y$

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