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Math Help - Multiple Integrals

  1. #1
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    Multiple Integrals

    I've done this problem, however not in the way suggested (which I assume is much quicker and easier!)

    By evaluating an appropriate double integral, find the volume of the wedge lying between the planes z=px and z = qx ( p > q > 0) and the cylinder x^2 + y^2 = 2ax
    (where a > 0)

    I did it by switching to cylindrical polars and then doing the triple integral as I couldnt see a method using a double integral to get the answer V=\pi (p-q)a^3

    Using the integral

    \int_{\phi=-\pi/2}^{\pi/2} \int_{r=0}^{2a\cos \phi} \int_{z=qr \cos \phi}^{pr \cos \phi} r dz dr d \phi

    any help with doing this with a better method?
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  2. #2
    MHF Contributor

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    so we also have that z \geq 0. first find the volume lying between each plane and the cylinder and then subtract the result. the volume lying between the plane z=px and the cylinder is:

    I(p)=\int \int_R px \ dy dx, where R: \ x^2 + y^2 \leq 2ax, \ 0 \leq x \leq 2a, which in polar coordinates becomes: I(p)=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \int_0^{2a \cos \theta} pr^2 \cos \theta \ d r d \theta=\pi p a^3. therefore: I(q)=\pi q a^3, and thus:

    V=I(p)-I(q)=\pi(p-q)a^3. \ \Box
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