Set up the triple integral for region between cylinders $x^2 + y^2 = 9 \quad x^2 + y^2 = 16 \quad z = 4+x^2$ and $0xy$ plane

Question

I ran into a problem that I am not sure about the correct answer. The question is:

Set up the triple integral for region between $x^2 + y^2 = 9 \quad x^2 + y^2 = 16 \quad z = 4+x^2$ and $0xy$ plane

What I have so far, using cylindrical coordinates, is

$$ 3 \leq \rho \leq 4\\ 0 \leq \theta \leq 2\pi\\ 0 \leq z \leq 4+(\rho cos \theta)^2 $$

How correct is my answer?

Answer 1

Seems good. The p between three and four sets a hollow cylinder of thickness one along the z axis, which is exactly what the first two equations prescribe. As for the z=4+x^2, simply plugging in pcostheta works in this case because the function never intersects the xy plane or anything else that would cause complications. As for the 0 to 2pi, that is of course correct because nothing is limiting the cylinder to one hemisphere or the other, for example a square root eliminating negative values. As far as I'm concerned, your bounds are completely correct.

Answer 2

How correct is my answer?

Looks good to me. Now just do

$$V = \iiint \rho d \rho d \theta dz$$

with those limits of integration.