Hi guys,
Would appreciate some help in explaining how to evaluate $\displaystyle \int\sqrt{4{r}^2-{r}^4}\, dr$.
1.
$\displaystyle \int\sqrt{4{r}^2-{r}^4}\, dr = \int(|r| \cdot \sqrt{4-{r}^2})\, dr = -{\frac12}\int(-2|r| \cdot \sqrt{4-{r}^2})\, dr$
2. You have to consider two different cases: r is positive or zero or r is negative.
3. Use now integration by substitution with $\displaystyle u = 4-r^2$.