# Tricky Spherical coordinates...

#### zollen

Consider the region S inside the cylinder x^2+y^2 = 1, outside the sphere x^2+y^2+z^2 = 1, and with z between 0 and 1. A view from above and a cross section (looking along the positive x axis) are shown below.

Set up one or more triple integrals to integrate a function f(x; y; z) over S using spherical coordinates, but DO NOT EVALUATE.

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#### zollen

$$\displaystyle \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{1}^{sec \phi } \rho^2 sin \phi d \rho d \phi d \theta + \int_{0}^{2 \pi} \int_{\pi / 4}^{\pi / 2} \int_{1}^{csc \phi } \rho^2 sin \phi d \rho d \phi d \theta$$
I guess I don't understand why the first time inner upper limit is $$\displaystyle sec \phi$$ while the second term inner upper limit is $$\displaystyle csc \phi$$
Both terms upper limits should have been $$\displaystyle sec \phi$$.
I don't understand why the second term upper limit is $$\displaystyle csc \phi$$