I'm trying to calculate the volume of an ellipsoid in cylindrical coordinates but it somehow doesn't turn out right.
The equation of my ellipsoid in cartesian coordinates is : $$ \left(\frac{x}{b}\right)^2+\left(\frac{y}{a}\right)^2+\left(\frac{z-c}{c}\right)^2=1 $$ which will be written as: $$ \left(\frac{r\cos(\theta)}{b}\right)^2+\left(\frac{r\sin(\theta)}{a}\right)^2+\left(\frac{z-c}{c}\right)^2=1 $$ in cylindrical coordinates. The following integral should give the volume of the ellipsoid: $$ \int^{2\pi}_{0}{d\theta}\int^{2c}_{0}{dz}\int^{r_{max(z,\theta)}}_{0}{rdr} $$ Where $r_{max(z,\theta)}$ is the biggest r for every fixed $z$ and $\theta$ which is: $$ \dfrac{ab\sqrt{1-\left(\dfrac{z-c}{c}\right)^2}}{\sqrt{a^2\cos^2{\theta}+b^2\sin^2{\theta}}} $$ Integrating this I get to the following integral which can't be solved in these limits: $$ \int^{2\pi}_{0}\frac{1}{a^2\cos^2{\theta}+b^2\sin^2{\theta}}{d\theta} $$ Can anyone help me understand what I'm doing wrong or send me the correct solution? I would really appreciate that! Please consider that I want to calculate the volume in cylindrical coordinates not spherical or cartesian.
Thank you very much.
HINT
Let use modified cylindrical coordinates
and $dx\,dy\,dz=abc \,r \,dr\, d\theta\, dz$
The ellipsoid equation becomes $r^2+z^2=1$ and the integral set up is
$$\int_0^{2\pi} d\theta \int_{-1}^{1} dz \int_0^{\sqrt{1-z^2}} abc \,r\,dr$$