I came across the integral a few days ago. Pretty interesting, many ways to solve this integral, be it graphically or by substitution.

$\int_a^b \sqrt{(x-a)(b-x)} dx =$?

Answer: $\pi \frac{(a-b)^2}{4}$

Hint: Consider the graph of $y = \sqrt{(x-a)(b-x)}$, it should give a semicircle which centre is $(\frac{a+b}{2},0)$