I have a rotationally-symmetric surface in $(r,z)$ (i.e. cylindrical coordinates) defined as follows: $$\frac{3r^2 - a}{(r^2 + z^2)^{3/2}} + \frac{3az^2}{(r^2+z^2)^{5/2}} - b = 0$$ where $a,b$ are real constants greater than zero, and $a<r^2$.
I'd like to find an expression for the volume enclosed by this surface, e.g. $V = 2\pi\int_0^R r z(r) dr$. However, the problem is that I can't separate the equation such that either $r$ or $z$ are isolated.
EDIT: One possibility that occurred to me was to integrate both sizes w.r.t. $z$, i.e.: $$\int\frac{3r^2 - a}{(r^2 + z^2)^{3/2}}dz + \int\frac{3az^2}{(r^2+z^2)^{5/2}}dz - \int b dz = 0$$ which would give: $$z\left[\frac{3(r^2+z^2)-a}{(r^2+z^2)^{3/2}}-b\right]=0$$
This is much easier to reorganise such that $r$ or $z$ are the subject, but is it "allowed"?
Is there are workaround for this, or will I have to resort to numerical methods?