I'm teaching a multivariable calculus course over the summer, and I stumbled upon an example problem from a prior instructor. It was presented in their notes right after the introduction of spherical coordinates, so I'm assuming they had intended it to be solved that way. However it is proving to be quite a hideous integral to set up.
The problem is as follows:
Integrate $f(x,y,z)$ over the region $W$, where $W$ is the region bounded between the spheres $x^2+y^2+z^2=1$ and $x^2+y^2+z^2=4$, with lower bound given by $z=x^2+y^2$, and only in the upper half space ($z\geq 0$.)
After screwing around with it for awhile, I'm come to think it's just a really bad intro example. I can't seem to set up the integral unless I break it up into two ridiculous pieces: $$ (a)\quad 0 \leq \phi \leq \arccos\left(\frac{-1+\sqrt{17}}{4}\right), \quad 1\leq \rho \leq 2 $$ $$ (b) \quad \arccos\left(\frac{-1+\sqrt{17}}{4}\right) \leq \phi \leq \arccos\left(\frac{-1+\sqrt{5}}{2}\right), \quad 1\leq \rho \leq \cot\phi \csc\phi $$
Does anyone see a better way? Or even see if I'm just going about it incorrectly, and making the wrong choice for order of integration?
I've taught multivariable calculus for most of my life (40+ years), and this is a horrendous region to do in spherical coordinates. From the fact that your predecessor has a redundant condition ($z\ge 0$), I infer that he/she intended the "lower bound" to be the cone $z = \sqrt{x^2+y^2}$ (i.e., the intended equation was $z^2=x^2+y^2$), and this makes for a wonderful spherical coordinates problem now.