Computing an intergral on a sphere I'm struggling with the following integral:
$$\int \limits_{S_t(x)} y_1^2 + y_2^2 + y_3^2 \, dA(y),$$
where $S_t(x) = \{y \in \mathbb{R}^3: \lvert y - x\rvert = t \}$.
I understand the integral above as average value of the integrand over $S_t(x)$. However I don't know how to compute it. 
I would appreciate any tips.

Translating the integral we get that
$$\int_{S_t(x)} y^2\:dA(y) = \int_{S_t(0)} (y+x)^2 \:dA(y)$$
Then use the fact that $dA(y) = t^2d\Omega$ 
$$\int_{S_t(0)} y^2+2y\cdot x + x^2 \:dA(y) = \int_{S^2} t^4 + x^2t^2\:d\Omega + \int_{S_t(0)} 2y\cdot x \:dA(y) = 4\pi t^2(x^2+t^2)$$
where the second integral vanished because the integrand was odd.
