I guess you know that the result can be written as a polynomial.
$$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$
So your question is how to estmate the coefficients.

 - $w_0$ is the volume, 
 - $w_n$ is the volume of unit ball.
 - $w_1$ is the area of the boundary, I think there is no good formula in general, but estimates are known and you could write it as an integral.
 - I am sure that there is no good formula for the rest of $w_i$.

If you want to write $w_i$ as an integral, 
check for example Burago--Zalgaller, Geometric inequalities.