Find the mass of the surface: $S=\{(x,y,z)| x^2+y^2\le 2x, z=\sqrt{x^2+y^2} \}$.
With mass density: $f(x,y,z)=x^2+y^2+z^2$.
So from the question I understand that I need to calculate $\iint_Sf(x,y,z)dS$.
What I did was I took this parametrization: $\vec S(t,r)=(1+r\cos(t), r\sin(t), r)$.
And found the normal vector length $|\vec S_r\times \vec S_t|$ and started calculating the integral with so much variables and alot of space to make lots of mistakes, and I got stuck integrating.
I want to know if there's any other hidden ways to solve this integral, I really wanted to use gauss/stokes theorems, but it doesn't work in this type of integral, but I'm positive there's something I'm missing that could've made this easier.
Any ideas and help is really appreciated, thanks in advance!
As you are integrating over the surface of the cone, parametrize the surface as,
$r(\rho, t) = (\rho \cos t, \rho \sin t, \rho)$
$|r_{\rho} \times r_t| = \rho \sqrt2$
Now note that $x^2 + y^2 \leq 2x \implies \rho \leq 2 \cos t, -\frac{\pi}{2} \leq t \leq \frac{\pi}{2}$
So the integral is,
$\displaystyle \int_{-\pi/2}^{\pi/2} \int_0^{2\cos t} \rho \sqrt2 \ (2 \rho^2) \ d\rho \ dt = 3 \sqrt2 \pi$