bounded by the paraboloid z=4x^2+4y^2 and the plane z=a (a>0) if S has constant density K.
I just need someone to explain to me how to set up the integral using cylindral coordinates.
We first note that $\displaystyle x_M=y_M=0$. To find $\displaystyle z_M$ using cylindrical coordinates, we let
$\displaystyle \begin{aligned}
x&=r\cos\theta\\
y&=r\sin\theta\\
z&=z.
\end{aligned}$
Our paraboloid equation becomes
$\displaystyle z=4r^2,$
and the equation of intersection of the paraboloid with the plane $\displaystyle z=a$ becomes
$\displaystyle \begin{aligned}
4r^2&=a\\
r&=\frac{\sqrt{a}}{2}.
\end{aligned}$
Our integral becomes
$\displaystyle z_M=\frac{1}{M}\int_0^{2\pi}\int_0^{\frac{\sqrt{a} }{2}}\int_0^a zKr\,dz\,dr\,d\theta,$
where
$\displaystyle M=\int_0^{2\pi}\int_0^{\frac{\sqrt{a}}{2}}\int_0^a Kr\,dz\,dr\,d\theta.$