I need to calculate volume of a figure:
$$\bigg(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\bigg)^2 = \frac{x}{h} + \frac{y}{k}, \quad x\geq 0, y \geq 0, z\geq 0$$
I switched it to cylindrical coordinate system:
$$x = a\cdot r\cdot \cos(\varphi)$$
$$y = a\cdot r\cdot \sin(\varphi)$$
$$z = z$$
$$r = \frac{a\cdot k\cdot \cos(\varphi ) + b\cdot h\cdot \sin(\varphi )}{h\cdot k(\cos(\varphi )+\sin(\varphi ))^2}$$
$$z = c\bigg[\sqrt{\frac{a\cdot r\cdot \cos(\varphi )}{h}+\frac{b\cdot r\cdot \sin(\varphi )}{k}}-r\big(\cos(\varphi )+\sin(\varphi )\big)\bigg]$$
$$ \text{Volume} =\int_{0}^{\pi/2}d\varphi\int_{0}^{r}rdr\int_{0}^{z}dz$$
At the end I've got this:
$$\frac{2}{5}\int_{0}^{\pi/2}\frac{(a\cdot k\cdot \cos(\varphi )+b\cdot h\cdot \sin(\varphi ))^3}{(h\cdot k)^3\cdot (\cos\varphi +\sin(\varphi ))^5}d\varphi$$
I can't solve this integral, maybe there is a mistake somewhere. Please tell me what to do next, or maybe there is another way to find the volume in my problem. Thank you!