Volume of solids of revolution

The volume ${\bf V = \pi * \int_{0}^{b} (a - x)^2 dy = }$

$$

${\bf \frac{a^2}{b^2}\pi * \int_{0}^{b} 2b^2 - y^2 -2b\sqrt{b^2 - y^2} dy = }$ $$

${\bf \frac{a^2}{b^2}\pi * (2b^2 y - \frac{y^3}{3})|_{0}^{b} - 2b \int_{0}^{b} \sqrt{b^2 - y^2} dy = }$ $$

${\bf \frac{5}{3} a^2 b \pi * - \frac{2 a^2\pi}{b} * \int_{0}^{b} \sqrt{b^2 - y^2} dy}$ $$

For ${\bf I = \int_{0}^{b} \sqrt{b^2 - y^2} }$ $$

Let ${\bf y = b sin\theta \implies dy = b cos\theta d\theta \implies }$ $$

${\bf I =\frac{b^2}{2} \int_{0}^{\frac{\pi}{2}} (1 + cos(2\theta)) d\theta = }$

${\bf \frac{b^2}{2} (\theta + \frac{1}{2} sin(2\theta))|_{0}^{\frac{\pi}{2}} = }$ ${\bf \frac{\pi}{4} b^2 \implies }$ $$

${\bf V(a,b) = \frac{5}{3} a^2 b \pi - \frac{a^2 b \pi^2}{2} = }$ ${\bf a^2 b \pi * (\frac{5}{3} - \frac{\pi}{2}) }$ ${\bf = a^2b\pi * (\frac{m}{n} - \frac{\pi}{p}) }$ $$

${\bf \implies m + n + p = 10. }$