Question: Calculate the volume of the cylinder $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ between the planes $z = 0$ and $z = 1 - \frac{x}{a}$.
Parametrising the solid, we have that $$ x = a\sin(\varphi) \ , \ y = b\cos(\varphi) \ , \ z = 1 - \sin(\varphi)$$ where $\varphi \in [0,2\pi]$, how can one use this parametrisation from here to find the volume of the cylinder?
Hint: check the following (assuming $\;a>0\,$):
$$\int_{-a}^a\int_{-\frac ba\sqrt{a^2-x^2}}^{\frac ba\sqrt{a^2-x^2}}\int_0^{1-x/a}dzdydx$$
...and perhaps changing to cylindrical coordinates can help.