Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$.
This is an exercise from my professor guide.
What I tried so far:
These exercise is obviously easier using spherical coordinates, with that in mind I've come up with the following limits:
$ 3 \le \rho \le 4 \\ 0 \le \phi \le \pi \\ 0 \le \theta\ \le \pi $
That is, the sphere cap formed by the 2 semi-spheres, more or less the volume described between the 2 surfaces in the image:
With the limits already established, I constructed the following integral:
$\int_0^\pi \int_0^\pi \int_3^4 \, \cos^2{(\theta)} \sin^3{(\phi)} \, \rho^4 \,\, \mathrm{d}\rho \, \mathrm{d}\phi \, \mathrm{d}\theta$ $= \frac{1562}{15}\pi$
However, the solution tha appears in the guide says that the result is:
$\frac{3124}{15}\pi$
Is there anything I'm doing wrong? I checked the exercise twice and I get the correct result IF I set the limits of $\theta$ from $0$ to $2\pi$, but that is the whole spherical surface not just the half described by the exercise.
Best regards, Daniel Rivas.
You did not get the right $\theta$! Here is what it should be
since the region in the $xz$-plane is an annulus.