# How to calculate the integral $\int_{S_{n-1}} x_1x_2dS$

How do I calculate the integral $$\int_{S^{n-1}} x_1x_2 dS$$ ,where $$S^{n-1}$$ is the $$n$$-dimensional sphere. The answer can have the expression $$\text{Vol} (S^{n-1})$$.

I know how to calculate the same integral when we are looking at the $$2$$-dimensional sphere or the $$3$$-dimensional sphere, since I can use the parametrization of those sphere and use the formula:

$$\int_{M} f = \int f \circ r(\theta, \alpha) \sqrt{\Gamma (r_{\theta}, r_{\alpha})}$$

Where $$\Gamma (r_{\theta}, r_{\alpha})$$ is the Gram determinant.

How can I do this in $$n$$-dimensions?

Help would be appreciated

• You could parametrize the "upper half" of $S^{n-1}$ by $\phi : B_{n-1}\to S^{n-1}$, $\phi(y) = (y,1-|y|^2)$, where $B_{n-1}$ is the $(n-1)$-dimensional ball in $\mathbb R^{n-1}$. – amsmath Apr 16 at 17:55
• I am not sure how that helps me – Gabi G Apr 16 at 20:53
• The area element at $(x_1, x_2, \ldots, x_n)$ and $(-x_1, x_2, \ldots, x_n)$ is the same, and the sphere is symmetric wrt the hyperplane $x_1 = 0$. – Maxim Apr 18 at 10:05
• Yes, this is how I eventually solved it – Gabi G Apr 18 at 10:43