I have a following integral: $$\int_0^1 dx\int_0^{\sqrt{1-x^2}}dy \int_\sqrt{x^2+y^2}^\sqrt{1-x^2-y^2}z^2dz$$
Which i have to solve by introducing polar coordinates, which is, by itself, relatively simple:
$$x=\rho\cos\theta\sin\phi \\ y=\rho\sin\theta\sin\phi \\ z=\rho\cos\phi$$
Besides this, i need to find Jacobian since i introduced a substitution, and since this is well known substitution Jacobian is $$J=\rho^2\sin\phi$$
Now, since i introduced polar coordinates, bounds of integral should be in polar form too, lower bound of the first, $dz$ integral, is simple $$\sqrt{x^2+y^2}= \rho\sin\phi$$, but i don't know what to do with this expression $$\sqrt{1-x^2}=\sqrt{1-\rho^2\cos^2\theta\sin^2\phi}$$
since, after introducing polar coordinates, this bound has all of the variables in itself, which makes it impossible to integrate over any of the variables i have, so i don't know how to solve this. Any help appreciated.