Calculate the nth moment given a joint density Suppose a point has a random location in the circle of radius 1 around the origin. The coordinates $(X,Y)$ of the point have a joint density 
$$f_{X,Y}(x,y) = \begin{cases}\frac{2}{\pi}(x^2+y^2)&\mathrm{\ if \ } x^2+y^2\le1\\ 0&\mathrm{\ otherwise\ }\end{cases}$$
Let $D$ be the distance from the random point to the center of the circle. How do I compute the $nth$ moment of $D$, $E(D^n)$, for $n = 1,2,...m$?

We have $D=\sqrt{X^2+Y^2}$, so the $n$-th moment of $D$ is the integral of
$$(x^2+y^2)^{n/2}\left(\frac{2}{\pi}\right)(x^2+y^2)$$
over the unit disk.
Thus we want to integrate $\displaystyle\frac{2}{\pi}\displaystyle(x^2+y^2)^{1+\frac{n}{2}}$ over the unit disk.
Change to polar coordinates. 
