the integral (from 0 to a) of sqrt(a^2 -r^2) 2r dr.
Thank you very much.
$\displaystyle 2\int_{0}^{a}r\sqrt{a^{2}-r^{2}}dr$
Let $\displaystyle r=asin{\theta}, \;\ dr=acos{\theta}d{\theta}$
$\displaystyle 2\int_{0}^{a}{asin{\theta}\sqrt{a^{2}(1-sin^{2}{\theta})}acos{\theta}}d{\theta}$
Remember, $\displaystyle 1-sin^{2}{\theta}=cos^{2}{\theta}$
This leads to:
$\displaystyle 2a^{3}\int_{0}^{a}{sin{\theta}cos^{2}{\theta}}d{\t heta}$
$\displaystyle -2a^{3}\frac{cos^{3}{\theta}}{3}$
But, $\displaystyle \theta=sin^{-1}(\frac{r}{a})$
Sub back in and get:
$\displaystyle \frac{2(r^{2}-a^{2})\sqrt{a^{2}-r^{2}}}{3}$
Sub in the limits of integration and get:
$\displaystyle 0-\frac{2a^{3}}{3}=\frac{2a^{3}}{3}$