Sunday

March 1, 2015

March 1, 2015

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.

**Answer this Question**

**Related Questions**

Calculus - Hi there I am having some troubles trying to do my calculus homework ...

Trig - Find the value of the six trigonometric functions. t = 7(pi)/4 Can you ...

Math 8R - help!!!!!!!!!!!!! - Multi-Step Equations Solving Equations with ...

algebra 2 - A. I need to write an equation to the linear equation..(-2,-2), (3,3...

Precalculus - Hello, Thanks for all of the help guys. I really appreciate it. I ...

Math: Calculus - Assume that a population size at time t is N(t) and that N(t)=...

Math - The following are the last two problems on my test review. I am trying to...

math 012 - Hi, I am trying to do homework for my math 012 class- solving linear ...

Math !!!!!!!! - Ok i have a Summer Math Assignment & i really need help on this...

Calculus - What is the simplest solution to the Brachistochrone problem and the ...