1
$\begingroup$

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.

  • 1
    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!2012-10-09
  • 0
    You are right. I had an explicit function in my mind but I thought there would be a formula...2012-10-09
  • 0
    How about you show us the explicit function?2012-10-09

1 Answers 1