I wish to calculate $\iiint_{E}z\sqrt{x^2+y^2}dV$ where $E$ is the domain trapped between the cylinder $x^2+y^2=2x$ and the planes $y=0$, $z=0$, $z=a$
In the answers sheet, the professor transforms the problem to cylinderical coordinates and writes:
$x = r\cos \theta$, $y = r\sin \theta$, $z=z$, and the limits are $0<z<a$, $0<\theta < \frac{\pi}{2}$, and $0<r<2\cos \theta$
I'm confused as to why $r < 2\cos \theta$. Shouldn't it be $r < \sqrt{2\cos \theta}$?
The points within the cylinder satisfy $x^2 + y^2 \leq 2x$. With the appropriate coordinate substitutions, $$r^2\cos^2\theta + r^2 \sin^2\theta \leq 2r\cos\theta, \implies r^2\leq 2r\cos\theta, \implies r\leq 2\cos\theta. $$ Then, to solve the integral, you need to expand $dV = rdrd\theta dz$ (don't forget the Jacobian $r$ factor!), make the appropriate substitutions within the integrand, and set the bounds $r \in [0,2\cos\theta]$, $\theta \in [0,\pi/2]$, $z \in [0,a]$. Can you go on from here?