# Math Help - Triple integrals

1. ## Triple integrals

1. The problem statement, all variables and given/known data

I have this question about triple integrals and spherical coordinates

2. Relevant equations

y = $\rho$ sin $\varphi$ sin $\theta$
x = $\rho$ sin $\varphi$ cos $\theta$
z = $\rho$ cos $\varphi$
$\rho$^2 = z^2 + y^2 + x^2

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

3. The attempt at a solution

I used the limits for the z to obtain z^2.
Thus, z^2 + x^2+y^2 = 4
Using the identity for $\rho$^2 = z^2 + y^2 + x^2 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 = $\sqrt{4-y^2}$ then x^2 + 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?

Since z^2^ +y^2 + x^2 = 4 is a sphere and spheres have a $\phi$ from 0 to $\pi$. Can I use those limits for $\phi$. Can anybody double check that my limits of integration are correct?

Thank you!

2. Originally Posted by jualin
1. The problem statement, all variables and given/known data

I have this question about triple integrals and spherical coordinates

2. Relevant equations

y = $\rho$ sin $\varphi$ sin $\theta$
x = $\rho$ sin $\varphi$ cos $\theta$
z = $\rho$ cos $\varphi$
$\rho$^2 = z^2 + y^2 + x^2

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

3. The attempt at a solution

I used the limits for the z to obtain z^2.
Thus, z^2 + x^2+y^2 = 4
Using the identity for $\rho$^2 = z^2 + y^2 + x^2 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 = $\sqrt{4-y^2}$ then x^2 + 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?

Since z^2^ +y^2 + x^2 = 4 is a sphere and spheres have a $\phi$ from 0 to $\pi$. Can I use those limits for $\phi$. Can anybody double check that my limits of integration are correct?

Thank you!

I tried 4 times to open the attached imageshack but it won't. Sorry.

Tonio

3. ## re:

Here is the link again. Imageshack - 81255254.

Or maybe this one will work