How can I calculate the Volume of $M = \left\{ (x,y,z)\in \mathbb R^3: 0\le z \le \sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\ \mathrm{and}\ \frac{x^2}{a^2} + \frac{y^2}{b^2} \le \sqrt{\frac{x}{a}} \right\}$ for $a,b >0$.
Aparently $M$ describes a half of some kind of elliptical cone.
1) I tried to start with generalized cylindrical coordinates: $\Phi(r, \phi, z) = \begin{pmatrix}ar \cos\phi\\br \sin\phi\\z\end{pmatrix}$
This gives $M=\left\{(r,\phi,z)\in \mathbb R^3: 0 \le z \le r\ \ \mathrm{and}\ r^2 \le \sqrt{r \cos\phi}\right\}$.
$r^2 \le \sqrt{r \cos\phi}$ means $\phi\in[-\pi/2,\pi/2]$ and $0 < r \le 1$.
The next step is to calculate a new surface element?
2) I also tried to do it in cartesian coords by resolving the y-radius of the Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} \le \sqrt{\frac{x}{a}} \Leftrightarrow y^2 \le \left[\left(\frac{x}{a}\right)^{1/2} - \left(\frac{x}{a}\right)^2 \right] b^2$. If I plot this function I can see the the right right is positive for $x\in[0,a]$ and hence $\left|y\right| \le \pm ab^2$.
Then I could calc $Vol(M) = \int_M dV = \int_{-ab^2}^{+ab^2} dy \int_0^a dx \int_0^{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}} dz$.
Question: Is this correct? Is there a easier or more elegant way to this? Is there an easy way to visualize M without plotting it on a computer?
Thanks!
EDIT: Corrected description: Before I asked for a spheroid.