The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder
and inside the sphere. The triple spherical integral for this problem is (from the answer key)
$$\int _0^{2\pi }\int _{\frac{\pi }{6}}^{\frac{5\pi }{6}}\int _{csc\phi }^2\:\rho ^2sin\phi \:d\rho \:d\phi \:d\theta $$
What is confusing me here is that there's some space at the endcaps of the cylinder that is not being accounted for. Why is this the case?
Best Answer
turns out that I was viewing the figures in the wrong way. As Andrew D. Hwang pointed out the cylinder has infinite height. Since the cylinder ends are open, it engulfs the top and bottom end of the sphere so the endcaps are not included. This explains why the integral:
$$\int _0^{2\pi }\int _{\frac{\pi }{6}}^{\frac{5\pi }{6}}\int _{csc\phi }^2\:\rho ^2sin\phi \:d\rho \:d\phi \:d\theta $$
is correct for this problem.