Let $B$ represent a sphere in 3D with radius $R$ and with center $x_0\in\mathbb{R}^3$ with $x_0\neq 0$. I would like to evaluate the following integral $$ I = \int_{\partial B} {x_1}^2 dS(x). $$ 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. First consider the case of a sphere at the origin. We have $$ \int_{\partial B}x_1^2 + x_2^2 + x_3^2 dS(x) = \int_{\partial B}|x|^2 dS(x) = \int_{\partial B}R^2 dS(x) = 4\pi R^4, $$ and then due we can obtain the value of $I \int_{\partial B} x_1^2 dx$ by dividing by $3$ due to symmetry.
Now if $B$ is not the origin the symmetry doesn't hold, and $|x|$ isn't a constant over the surface of the sphere.
So is there any to obtain an analytic solution in this case?
Let ${\bf p}=(p_1,p_2,p_3)$ be the center of your sphere and introduce new coordinates $\bar x_i$ by $$x_i=p_i+\bar x_i\qquad(1\leq i\leq3)\ .$$ Then your integral becomes $$I=\int_{\partial\bar B}(p_1+\bar x_1)^2\>{\rm d}S(\bar {\bf x})\ .$$ Here $\bar B$ is the ball of radius $R$ centered at the origin of $\bar{\bf x}$-space. The rest is easy.