I'm trying to calculate the three-dimensional Lebesgue-measure of a set bounded by two spheres in $\mathbb{R}^3$:
$x^2+y^2+z^2-2z =0;$ $x^2+y^2+z^2 =2;$
I know I should use cylindrical coordinates for substitution, I just can't seem to figure out how to get the bounds of the integrals and what the order of the integrals should be. Secondly, does this use the Fubini theorem and what is the correct argumentation for it's use?
We seek to find the following volume$$V=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{2\cos\theta}^{\sqrt 2}r^2\sin\theta\,dr\,d\theta\,d\phi=\int_{0}^{2\pi}\int_{0}^{\pi}\dfrac{1}{3}(2\sqrt 2-8\cos^3\theta)\sin\theta\,d\theta\,d\phi=\dfrac{8\pi(\sqrt 2-1)}{3}$$