Triple Integration in vector calc NIn the following example I have been asked to find the volume V of the solid bounded by the sphere $ x^2 + y^2 +z^2 = 2 $ and the paraboloid $ x^2 + y^2 = z $ by using triple integration.
I am not quite sure how to set up this triple integration and what each of the integrands should be.

It is comfortable to describe your solid in cylindrical coordinates since both the sphere and the paraboloid have the $z$-axis as an axis of symmetry (both surfaces are surfaces of revolution around the $z$-axis). 
In cylindrical coordinates, the sphere is described by $\rho^2 + z^2 = 2$ and the paraboloid is described by $\rho^2 = z$. The solid bounded between them is described by $\rho^2 \leq z \leq \sqrt{2 - \rho^2}$ and so
$$ V = \int_{0}^1 \int_{0}^{2\pi} \int_{\rho^2}^{\sqrt{2 - \rho^2}} \rho \, dz \, d\theta \, d\rho = 2\pi \int_0^{1} \rho \left( \sqrt{2 - \rho^2} - \rho^2 \right) \, d\rho = 
2\pi \left( \frac{1}{2} \int_1^2 \sqrt{u} \, du - \int_0^{1} \rho^3 \right)
=  2\pi \left( \frac{2^{\frac{3}{2}} - 1}{3} - \frac{1}{4}\right).$$
where we used the substitution $u = 2 - \rho^2, \, du = -2\rho$ for the first integral.
