Volume enclosed by a cone in spherical coordinates

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March 12th 2009, 07:30 PM
s7b

Volume enclosed by a cone in spherical coordinates

Hi,

I really don't understand how to set up this integral;

Find the volume of the solid enclosed by the cone z=sqrt(x^2 + y^2) between the planes z=1 and z=2

The only thing I did so far was convert the first z equation into phi=pi/4
I'm not sure where to go from here.
March 12th 2009, 09:15 PM
ThePerfectHacker

Quote:

Originally Posted by s7b

Hi,

I really don't understand how to set up this integral;

Find the volume of the solid enclosed by the cone z=sqrt(x^2 + y^2) between the planes z=1 and z=2

The only thing I did so far was convert the first z equation into phi=pi/4
I'm not sure where to go from here.

The $\phi$ angle opens up from $0$ to $\tfrac{\pi}{4}$ .
The $\theta$ angles travels a complete revolution.
What is trickier is finding the bounds on $\rho$ .

If $z=1$ then the plane cut is $1$ unit above the $xy$ plane, but the side of the cone makes an angle of $\tfrac{\pi}{4}$ , therefore the length of the side of the cone is $\cos \tfrac{\pi}{4}=\tfrac{\sqrt{2}}{2}$ . For $z=2$ by similar argument we get $\rho = 2\cdot \tfrac{\sqrt{2}}{2} = \sqrt{2}$ . Therefore, the integral becomes,
$\int_{\sqrt{2}/2}^{\sqrt{2}} \int_0^{2\pi} \int_0^{\pi/4}\rho^2 \sin \phi ~ d\phi ~ d\theta ~ d\rho$
April 11th 2009, 06:07 AM
fisix

Quote:

Originally Posted by ThePerfectHacker

The $\phi$ angle opens up from $0$ to $\tfrac{\pi}{4}$ .
The $\theta$ angles travels a complete revolution.
What is trickier is finding the bounds on $\rho$ .

If $z=1$ then the plane cut is $1$ unit above the $xy$ plane, but the side of the cone makes an angle of $\tfrac{\pi}{4}$ , therefore the length of the side of the cone is $\cos \tfrac{\pi}{4}=\tfrac{\sqrt{2}}{2}$ . For $z=2$ by similar argument we get $\rho = 2\cdot \tfrac{\sqrt{2}}{2} = \sqrt{2}$ . Therefore, the integral becomes,
$\int_{\sqrt{2}/2}^{\sqrt{2}} \int_0^{2\pi} \int_0^{\pi/4}\rho^2 \sin \phi ~ d\phi ~ d\theta ~ d\rho$

Surely the limits for r would be from $r=\sqrt{2}$ to $r=2\sqrt{2}$ . No? Using the lengths as $\tfrac{1}{\cos \tfrac{\pi}{4}}$ and $\tfrac{2}{\cos \tfrac{\pi}{4}}$ .