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.

Alternative solution:
With substitutions $y_1 = x_1 + t\sin \phi \cos \theta$,
$y_2 = x_2 + t\sin \phi \sin \theta$ and $y_3 = x_3 + t\cos \phi$
for $\phi \in [0, \pi]$ and $\theta \in [0, 2\pi)$.
Then $\mathrm{d} A(y) = t^2 \sin\phi \ \mathrm{d}\phi \ \mathrm{d}\theta$.
See: http://math.mit.edu/~jorloff/suppnotes/suppnotes02/v9.pdf
We have
\begin{align}
I &= \int_0^\pi \int_0^{2\pi} 
[(x_1 + t\sin \phi \ \cos \theta)^2 + (x_2 + t\sin \phi\ \sin \theta)^2 + (x_3 + t\cos \phi)^2]
t^2 \sin\phi \ \mathrm{d}\theta \ \mathrm{d}\phi\\
&= 4\pi t^2 (x_1^2 + x_2^2 + x_3^2 + t^2).
\end{align}
