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

I have this question about triple integrals and spherical coordinates

2. Relevant equations

y = $\displaystyle \rho$ sin $\displaystyle \varphi$ sin $\displaystyle \theta$

x = $\displaystyle \rho$ sin $\displaystyle \varphi$ cos $\displaystyle \theta$

z = $\displaystyle \rho$ cos $\displaystyle \varphi$

$\displaystyle \rho$^2 = z^2 + y^2 + x^2

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

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 $\displaystyle \rho$^2 = z^2 + y^2 + x^2 then $\displaystyle \rho$^2 = 4

which gives me a value of $\displaystyle \rho$ = 2.

To get $\displaystyle \theta$ I graphed the x limits of the integral. Since x = $\displaystyle \sqrt{4-y^2}$ then x^2 + y ^2 =4. Therefore it is a circle of radius 2. Thus I assumed that $\displaystyle \theta$ goes from 0 to 2$\displaystyle \pi$.

Now my problem is to find the limits for $\displaystyle \varphi$ which I don't know how to get.

Any ideas on how to solve for $\displaystyle \varphi$ 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 $\displaystyle \phi$ from 0 to $\displaystyle \pi$. Can I use those limits for $\displaystyle \phi$. Can anybody double check that my limits of integration are correct?

Thank you!