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.

Printable View

- Nov 11th 2009, 11:49 AMdaskywalkerFind 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. - Nov 11th 2009, 04:39 PMScott H
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

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