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Math Help - Integral Coordinate Conversion

  1. #1
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    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.
    Attached Thumbnails Attached Thumbnails Integral Coordinate Conversion-integral-conversion.jpg   Integral Coordinate Conversion-sample-1.jpg  
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  2. #2
    Eater of Worlds
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    For the first part.

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

    This is a cylinder of height 8 and radius 1.

    In rectangular:

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