Triple Integrals with Spherical Coordinates: Finding Limits

[jualin]

Homework Statement

I have this question about triple integrals and spherical coordinates

http://img405.imageshack.us/img405/9343/81255254.th.jpg

Homework Equations

y = \rho sin \varphi sin \theta
x = \rho sin \varphi cos \theta
z = \rho cos \varphi
\rho² = z² + y² + x²

This is the way
http://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords_files/eq0007MP.gif"

Thus I need to find the limits of integration for \rho \theta and \varphi

The Attempt at a Solution

I used the limits for the z to obtain z².
Thus, z² + x² +y² = 4
Using the identity for \rho² = z² + y² + x² then \rho² = 4
which gives me a value of \rho = 2.

To get \theta I graphed the x limits of the integral. Since x = \sqrt{4-y²} then x² + y ² =4. Therefore it is a circle of radius 2. Thus I assumed that \theta goes from 0 to 2\pi.
Now my problem is to find the limits for \varphi which I don't know how to get.

Any ideas on how to solve for \varphi and also, can someone double check that the other limits of integration are correct?

Thank you!

 

[rado5]

How do you derive the spherical coordinates? You can find the ranges of \phi in the definition of spherical coordinates, so study your book again! And the ranges for \rho and \theta are correct.

 

[jualin]

How do you derive the spherical coordinates? You can find the ranges of \phi in the definition of spherical coordinates, so study your book again! And the ranges for \rho and \theta are correct.

Can I use the limits of y to get \phi. For instance since y = 4 then can I say
\rho sin \phi sin \theta = 4 so sin \phi = \rho / sin \theta

Now I am stuck there. Do I plug in a value for \rho and \theta. For instance 2 for \rho and 2pi for \theta. That would give me an undefined answer and sin \phi is always defined. Where do I go from here?
Thank you for the quick response

 

[jualin]

Or since z² +y² + x² = 4 is a sphere and spheres have a \phi from 0 to \pi. Can anybody double check that my limits of integration are correct?

Thank you

 

[rado5]

Yes your bounderies are now correct and I'm sure about that, because I have a similar example in my book with answer.

\rho is between 0 to 2 and \theta is between 0 to 2\pi and \phi is between 0 to \pi