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April 24, 2014

April 24, 2014

Posted by **shylo** on Wednesday, November 25, 2009 at 9:24am.

Here is the question:

Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.

- Calculus -
**Count Iblis**, Wednesday, November 25, 2009 at 10:27amWhat you have to do here is show that the volume element:

dxdydz can be written as

r^2 sin(theta)dphi dtheta dr

where theta is the angle w.r.t. the z-axis and phi is the angle that corresponds to rotating around the z-axis.

It is easy to see that this is the volume element because you can see the three orthogonal length elements hee:

r dtheta

r sin(theta) dphi

Note that if you rotate around the z-axis, your radius will be

r sin(theta)

and

dr

If you want to prove this formally by direct substituton of

x = r sin(theta)cos(phi)

y = r sin(theta)sin(phi)

z = r cos(theta)

You have to write down the Jacobian, i.e. the 3x3 matrix of partial deivatives of the the three cartesian coordinates w.r.t. r, theta and phi.

Once you've got that the volume element is r^2 sin(theta)dphi dtheta dr you can integrate this straightforwadly. r ranges from zero to R, phi goes from zero to 2 pi and theta goes from zero to pi.

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