Find the volume of the given solid.
Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant
Because of the obvious cylindrical symmetry, I would recommend changing to polar coordinates in the yz-plane. Then $\displaystyle y= r cos(\theta)$, $\displaystyle z= r sin(\theta)$ and the cylinder, $\displaystyle y^2+ z^2= 4$ becomes r= 2 and x= 2y gives height $\displaystyle x= 2r cos(\theta)$.
Integrate $\displaystyle 2r cos(\theta) r dr d\theta$ with r ranging from 0 to 2 and $\displaystyle \theta$ ranging from 0 to $\displaystyle \pi/2$.
(Normally, of course, we use polar coordinates in an xy-plane. Essentially what I did was swap x and z so the problem would be "find the area of the solid bounded by $\displaystyle x^2+ y^2= 4$ in the first quadrant from z= 0 to z= 2y. Changing the coordinate system doesn't change the solid so the two solids have the same volume)