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.