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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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    How about you show us the explicit function?2012-10-09

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You can apply Fubini at this step \begin{align*} \int_0^r s^{-n}\int_{B_s} f(x)\;dx\,ds &= \int_0^r s^{-n}\int_0^s \int_{\partial B_t} f(x)\;dS(x)\,dt\,ds\\ &= \int_0^r \int_t^r s^{-n}\int_{\partial B_t} f(x)\;dS(x)\, ds\, dt\\ &= \int_0^r \int_{\partial B_t} f(x)\; dS(x) \cdot \int_t^r s^{-n}\, dt\, ds\\ &= \int_0^r \int_{\partial B_t} f(x)\; dS(x) \cdot \frac{t^{1-n} - r^{1-n}}{n-1}\, ds\\ &= \int_0^r \int_{\partial B_t} \left(\frac{|x|^{1-n} - r^{1-n}}{n-1}\right)f(x)\; dS(x)\, dt\\ &= \int_{B_r} \left(\frac{|x|^{1-n} - r^{1-n}}{n-1}\right)f(x)\; dx\, dt\\ \end{align*}

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    @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. $\int_0^r \int_t^r\cdots\; ds\, dt$.2012-10-09