In my course we had to calculate the inner volume of a paraboloid described with $f(x, y) = x^2 + y^2$ by converting cartesian coordinates to polar coordinates.
The formula I ended up with was $f(r, \psi) = r^2$. This was then used to calculate the Volume underneath the paraboloid using:
$V =\int_0^{2\pi}\int_0^R{r^2 *r} \operatorname{dr} \operatorname{d\psi} = \frac{2\pi}{4}[r^4]_0^R$
Now, I don't understand what happens if you plug in a value contaning a unit. I.e. if $R=2m$ wouldn't this give $V = 8\pi m^4$ ?