Multivariable Calculus Volume of Integration Question

Question

I have an integral $$ \iiint\sqrt{x^{2} + y^{2} + z^{2}\,}\,{\rm e}^{-\left(x^{2} + y^{2} + z^{2}\right)}\, {\rm d}x\,{\rm d}y\,{\rm d}z $$ The integration region is bounded by the sphere $\left\{\left(x,y,z\right)\ \ni\ \sqrt{x^{2} + y^{2} + z^{2}\,}\, < 4\right\}$. I so far have set up the integral in spherical coordinates subsituting the rho in all of the necessary places. The problem I am having is setting up the bounds for the spherical integral. I know that $0<\theta<2\pi$ and $0<\rho<2$ but I am not sure about how I would set up the final integral in terms of $\phi$. Thanks.

Answer 1

Usually $\phi \in [-\pi/2,\pi/2]$.

Answer 2

The answer depends on the parametrization you use to convert Cartesian into spherical coordinates: http://mathworld.wolfram.com/SphericalCoordinates.html

If we go with the parametrization $$(x,y,z) = (\rho \cos \theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi),$$ then we have $$\rho \in [0, \infty), \quad \theta \in [0, 2\pi), \quad \phi \in [0, \pi].$$ But as I pointed out, this is not the only parametrization. In all cases, you have to look at the mapping itself to get the appropriate intervals for each parameter.