How to evaluate $I = \int_{\partial B} {x_1}^2 dx$ where $B$ is a sphere in 3D?

Question

Let $B$ represent a sphere in 3D with radius $R$. I would like to evaluate the following integral $$ I = \int_{\partial B} {x_1}^2 dx. $$ That is, each point on the surface of the sphere is represented by the vector $x = (x_1,x_2,x_3)^\top$, and I want to integrate the square of the first component over the entire sphere. Can this be done analytically?

Answer 1

In this specific case, by symmetry, we can say that all $$\int_{\partial B}{x_i}^2dS(x)$$ are equal ($i=1,2,3$), but their sum is $R^2 vol(\partial B)=4\pi R^4$, so...