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Math Help - Question to do with volume of a solid between a paraboloid and a plane

  1. #1
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    Question to do with volume of a solid between a paraboloid and a plane

    Hi

    the question is

    Use polar coordinates to find the volume of the solid which is under the paraboloid z=x^{2}+y^{2}and above the disk x^{2}+y^{2}\leq9 in the plane z=0.

    thanks
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  2. #2
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    Aug 2008
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    If z=x^2+y^2 then z=f(r,t)=r^2 so we integrate over the circle x^2+y^2=9 as the radius goes from 0 to 3 and all the way round the circle or just 4 times the part in the first quadrant since f is symmetrical:

    V=4\int_0^{\pi/2}\int_0^3 r f(r,t) dr dt

    and you can check that by just calculating the volume inside the paraboloid via shells and then subtract that from just the cylinder it's housed in:

    \pi (3)^2 9-\pi\int_0^9 ydy=4\int_0^{\pi/2}\int_0^3 r f(r,t) dr dt

    half-in and half-out right?
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