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 $n = 1,2,...m$?

For every $0\leqslant x\leqslant1$, $\mathrm P(D\leqslant x)=\int\limits_0^{2\pi}\int\limits_0^x(2/\pi)r^2\cdot r\mathrm dr\mathrm d\theta=\int_0^x4r^3\mathrm dr$ hence the density $f_D$ of $D$ is such that $f_D(x)=4x^3\cdot \mathbf 1_{0\leqslant x\leqslant1}$.
In particular, $\mathrm E(D^n)=\int\limits_0^1x^n\cdot f_D(x)\mathrm dx=\int\limits_0^14x^{n+3}\mathrm dx=4/(n+4)$.
One can note that $D^4$ is uniform on $(0,1)$.
