# Math Help - Find mass and center of mass of the solid S....

1. ## Find mass and center of mass of the solid S....

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.

2. We first note that $x_M=y_M=0$. To find $z_M$ using cylindrical coordinates, we let

\begin{aligned}
x&=r\cos\theta\\
y&=r\sin\theta\\
z&=z.
\end{aligned}

Our paraboloid equation becomes

$z=4r^2,$

and the equation of intersection of the paraboloid with the plane $z=a$ becomes

\begin{aligned}
4r^2&=a\\
r&=\frac{\sqrt{a}}{2}.
\end{aligned}

Our integral becomes

$z_M=\frac{1}{M}\int_0^{2\pi}\int_0^{\frac{\sqrt{a} }{2}}\int_0^a zKr\,dz\,dr\,d\theta,$

where

$M=\int_0^{2\pi}\int_0^{\frac{\sqrt{a}}{2}}\int_0^a Kr\,dz\,dr\,d\theta.$