An example of computation of volume of body in $R^3$

Question

How would you solve the following exercise: Compute the volume of the body in $R^3$ bounded by the surface $(x^2+y^2)^2=z^2(3x^2+y^2)$ and planes $z=0$ and $z=5$.

Answer 1

With the cylindrical equation $r=z\sqrt{1+2\cos^2\theta}$, set up the volume integral as,

$$V= \int_0^{2\pi}d\theta\int_0^{5\sqrt{1+2\cos^2\theta}}\left(5-\frac r{\sqrt{1+2\cos^2\theta}}\right)rdr$$ $$=\int_0^{2\pi}\frac{125}6 (1+2\cos^2\theta) d\theta = \frac{250\pi}3$$

Answer 2

So far I agreed with achille hui's answer. We have $V=\int\int\int_{V}1dxdydz$, and so, after passing to cylindrical coordinates, we get $V=\int_0^5\int_0^{2\pi}\int_0^{z\sqrt{1+2\cos^2\theta}}\rho d\rho d\theta dz={{250}\over{3}}\pi$. This looks correct to me.