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Math Help - Double Integral with Polar Coordinates

  1. #1
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    Double Integral with Polar Coordinates

    Problem:

    Use polar coordinates to find the volume of the given solid.

    Enclosed by the hyperboloid

    -x^{2} - y^{2} + z^{2} = 1 and the plane z=2

    My attempt:

    I think I'm tripping up on the radius portion of this problem. I tried to solve the function for z and set the two bounds equal to find their intersection and therefore the radius. I substituted z for two while finding the radius, which is something I probably should not do. Any help would be great. Attached is an image of my attempt.

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  2. #2
    MHF Contributor
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    Hi

    The first thing to do is to draw a sketch.
    The plane z=2 is easy to draw.
    The hyperboloid is x+y=z-1. Now x+y being positive, z is higher than 1 (or lower than -1 which we do not consider here). For any value k of z (higher than 1) the intersection of the plane z=k with the hyperboloid is the circle x+y=k-1. And in polar coordinates x+y=r.

    The volume is therefore \int \int \int r dr d\theta dz. I let you find the boundaries for each variable.
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