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

**daskywalker** · November 11th 2009, 11:49 AM

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.

**Scott H** · November 11th 2009, 04:39 PM

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

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

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