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 $$\varphi$$
$$\rho$$² = 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 $$\rho$$² = z² + y² + x² then $$\rho$$² = 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 $$\pi$$