1. ## Integral Coordinate Conversion

Hello again. I am going over a test review and had a few more problems. The first question, I can do the spherical conversion with no problems, but I'm just having a hard time visualizing and doing rectangular.

The second question is a Green's Theorem problem. Using the theorem the integral becomes very very simple (1-x). Our professor actually did this problem in class--and switched to a polar coordinate system. At this point he introduced a rule/trick/something? playing on even and odd values of n that made the integration of sin^n and cos^n simple. I have some jumbled notes, but can't really connect the dots so I can reproduce what was done. Anyway I'm rambling and being vague so I understand if this doesn't make any sense.

2. For the first part.

$\displaystyle \int_{0}^{2\pi}\int_{0}^{1}\int_{-4}^{4}rdzdrd{\theta}$

This is a cylinder of height 8 and radius 1.

In rectangular:

$\displaystyle \int_{-1}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\int_{0}^{8}dzdydx$