finding volume bound by cylinder, cone, and xy plane

Question

I am trying to calculate the volume of a solid enclosed by the cone $z= \sqrt{(x^2+y^2)}$, the cylinder $(x+3/2)^2 + y^2 = 9/4$, and the plane $z=0$.

When I converted to cylindrical formulas I got $z=0 ,z=p ,p=-3\cos(\theta)$ but I am having troubles on the limits.

Answer 1

For a correct set up of the integral we need to make a sketch similar to this one here attached for $z-x/y$ and $x-y$ planes.

In cylindrical coordinates the region is determined by

• $0\le z \le \frac32$
• $\theta_{1}(z)\le \theta \le \theta_{2}(z)$
• $z\le r \le -\frac32 \cos \theta$

and for symmetry we can also consider twice the integral over the region

• $0\le z \le \frac32$
• $\theta_{1}(z)\le \theta \le \pi$
• $z\le r \le -\frac32 \cos \theta$

thus we only need to find $\theta_1(z)$ to obtain the correct set up.