Calculate $\int_{S^{n-1}} x_1x_2 dS$

Question

How do I calculate $\int_{S^{n-1}} x_1x_2 dS$ (where $S^{n-1}$ is the $n-1$ dimensional sphere in $\mathbb{R}^n$)?
At the first part of the question I needed to calculate $I=\int_{S^{n-1}} x_1^2$ and to do that I just used symmetry, to get $nI = \sum_{i=1}^{n} \int_{S^{n-1}} x_i^2 = \int_{S^{n-1}} 1$, but what do I do here? I am allowed to write the answer in terms of $\int_{S^{n-1}}1$. Thanks!

Answer 1

$\displaystyle\int_{S^{n-1}}x_1x_2 \, dS = 0$ by the change of variables formula. If you make the change of variables $(x_1, \dots, x_n) \mapsto (-x_1, x_2, \dots, x_n)$, the integrand picks up a $-$sign, while the region of integration stays the same.