I have a question to a pretty basic integration problem. I was pretty sure about my solution but my tutor had a different one such that I am confused now. The integral is the following:
$$\int_{S_r(0)} z^2 d S_r(0)$$
Using the standard parameterization, I obtain:
$$d S = r^2 \sin(\theta) d \theta d \phi$$
$$\int_{S_r(0)} z^2 d S_r(0) = \int_{0}^{2 \pi} \int_{0}^{\pi} r^2 \cos^2(\theta) r^2 \sin(\theta) d \theta d \phi = \dots = \frac{4 r^4\pi}{3}$$
I would be very glad if someone could verify or falsify the result!
Thanks much in advance
Andreas
$I=r^4\displaystyle \int_{0}^{2\pi} d\phi .\left(-\displaystyle \int_{0}^{\pi}cos^2\theta \ d(cos\theta)\right)$
$I=\dfrac{1}{3}r^4\displaystyle \int_{0}^{2\pi} d\phi .\left[cos^3\theta\right]_{\pi}^{0}$
$I=\dfrac{2}{3}r^4\displaystyle \int_{0}^{2\pi} d\phi $
$I=\left(\dfrac{2}{3}r^4.2\pi\right)=\dfrac{4\pi}{3}r^4 $