# Integral over a ball.

How can we calculate the following integral? $$\int_{0}^r\frac{1}{s^n}\int_{B(s)}f(x)dxds$$ Here $B(s)$ is the ball of radius $s$ centered at the origin.

I think that this can be computed by $$\int_{0}^r\frac{1}{s^n}\int_{B(s)}f(x)dxds =\int_{0}^r\frac{1}{s^n}\int_{0}^s\int_{\partial B(t)}f(x)dxdsdt$$ But I am stuck at this point. Any help is more than welcome.

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If you don't know what is $f$, there's no point in actually trying. This expression is perfectly suitable for arbitrary $f$ as "the value of the integral". If you want to compute it, work with an explicit function! – Patrick Da Silva Oct 9 '12 at 7:15
You are right. I had an explicit function in my mind but I thought there would be a formula... – Thom M Oct 9 '12 at 8:22
How about you show us the explicit function? – Patrick Da Silva Oct 9 '12 at 14:06

@ThomM I exchanged the order of integration. We are integrating over $\{(s,t) \mid 0 \le t \le s \le r\}$, right? There are to possibilities to integrate over this domain: First $t$, then $s$, i. e. $\int_0^r\int_0^s \cdots \; dt\,ds$ or first $s$ then $t$, i. e. $\int_0^r \int_t^r\cdots\; ds\, dt$. – martini Oct 9 '12 at 8:32