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Homework Help: A difficult multiple integral question

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data

    I don't have the answer for this one, so hopefully someone can help...

    (a) By reversing the order of integration, evaluate [tex]\int_0^8 \int_{y^{1/3}}^2 \sqrt{x^4 + 1} dx dy[/tex]

    (b) 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 [tex]x^2 + y^2 = 2ax[/tex] where a> 0.

    Find also the area of the curved surface of the wedge.

    3. The attempt at a solution

    This is a really strange, and in my opinion, incredibly difficult question. The first part is straightforward.

    For the second part, I tried to visualise the wedge in the x-z, x-y and y-z dimensions. I settled for a view in the x-z dimension:


    and then did a relative volume. The area enclosed between the lines is

    [tex]\int_0^{2a} \int_{qx}^{px} dy dx = 2a^2 (p-q)[/tex]

    then the relative volume is given by [tex]\pi a^3 (p-q)[/tex]

    Can somebody help check this please? I don't think it's the right answer, seeing as it has absolutely nothing to do with the (a) integral, which I'm guessing is a hint of some sort.

    How do I work out the second part? (Just some hints please.) I am thinking of parameterisation at the moment (since y is dependent on x, z is dependent on x), although not quite sure how to do that.
  2. jcsd
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