PDE Problem - (really just an integral)

This problem isn't too bad in theory. I understand it, I'm just having great difficulty expressing the answer, so I'm wondering if anyone has some method, or insight how to "see" or express the answer.

Question:

{

Find $\displaystyle u(r,t) $ from the formula:

$\displaystyle ru = \frac{1}{2c}\int_{r-t}^{r+t}sv_0(s)ds $

If $\displaystyle v_0=1 $ in the solid sphere $\displaystyle |r| \leq R $ and $\displaystyle v_0=0$ outside. ...

}

The "..." just represents the other part of the question that I'm not interested in.

As an example if we set:

$\displaystyle c=1 $

$\displaystyle R=8 $

$\displaystyle r=1 $

$\displaystyle t=10 $

$\displaystyle r -t =-9 $

$\displaystyle r+t=11$

The integral would drop to:

$\displaystyle \frac{1}{2}\int_{r-t}^{r+t}sv_0(s)ds=\frac{1}{2}\int_{-9}^{8}sds $

This makes sense, that as long as we integrate below R we get our integral back. But when we get above R then the integral returns 0. My question is this I guess...

Is there an easy way to setup all these equalities and integrals? Because I am getting lost when I try to do it. The only time it really makes sense is when I plug in numbers and force my way through it. I'm just curious what a better way to accomplish this task of setting up cases is. Thanks in advance.