Continuous random variables unit ball in $\mathbb R^n$

Question

I am currently having some difficulties solving this question...

If a point is chosen uniformly at random from the unit ball in $\mathbb{R}^{n}$ (that is, the set$$\{(x_1,\ldots,x_n) : x_1^2+\cdots+x_n^2\leq1\},$$and $L_n$ is the distance of the point from the origin, what is $E(L_n)$?

My attempt at the question

I know in the case for a unit disk, the distance is just $\sqrt{x^2+y^2}$, and the join density function would be $\frac{1}{\pi}$, and integration would be very easy in polar coordinates. However, I am not sure how to grasp this question.

Answer 1

Note that $F(r)=P[L \le r] = r^n$, and $F'(r) = n r^{n-1}$.

Hence $EL = \int_0^1 r F'(r)dr = {n \over n+1}$.