Calculus
posted by shylo on .
Hi there i am having some problems trying to do my calculus homework and i really need help on how to show the step to proof the volume of a sphere which is V= 4/3pirsquare. But I have to use triple integral to proof the volume of a sphere. Please help me and give me some good website that i can find the proper formula to use.
Here is the question:
Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r.

What 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 zaxis and phi is the angle that corresponds to rotating around the zaxis.
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 zaxis, 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.