I am using Kirchoff's formula to solve wave equation in three dimensions, and I have the following three integrals:
$\int_{\partial B(x_0,t)}t*|y-x_0|^2dS(y)$
$\int_{\partial B(x_0,t)}|y-x_0|^6dS(y)$
$\int_{\partial B(x_0,t)}\nabla|y-x_0|^6\cdot (y-x_0)dS(y)$
How can I get rid of the integrals? What formulas/theorems can I use to get out of them?
First translate all of the integrals to get
$$I_1 = \int_{\partial B(0,t)}t|y|^2\:dS(y)$$
$$I_2 = \int_{\partial B(0,t)}|y|^6\:dS(y)$$
$$I_3 = \int_{\partial B(0,t)}\nabla|y|\cdot y\:dS(y)$$
For the first two integrals, note that
$$I(n) = \int_{\partial B(0,t)} |y|^n \:dS(y) = \int_{S^2} t^{n+2} d\Omega = 4\pi t^{n+2}$$
meaning
$$I_1 = t\cdot I(2) = 4\pi t^5$$
$$I_2 = I(6) = 4\pi t^8$$
Lastly using that
$$\nabla |y| = \frac{y}{|y|}$$
we get that
$$I_3 = \int_{\partial B(0,t)} |y| \:dS(y) = I(1) = 4\pi t^3$$