I need to understand how the following integrals depend on the dimension $d$; the result should be about a (negative) power of $d$. Let $\mathbb{B}^d$ be the $d$-dimensional ball of radius $1$, centered on the origin. Let $|A|$ be the $d$-dimensional Lebesgue measure of $A$:
a) $$\frac{1}{|\mathbb{B}^d|}\int_{\mathbb{B}^d}dx_1...dx_d x_i^2 $$
b) $$\frac{1}{|\mathbb{B}^d|}\int_{\mathbb{B}^d}dx_1...dx_d x_i^2 x_j^2. $$
I receaved some hints about using recurrence results spheres and balls but I didn't succeded to overcome them till now. Thank you