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Math Help - VOLUME INTEGRATION HELP PLZ!

  1. #1
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    VOLUME INTEGRATION HELP PLZ!

    Q: Find the volume of the solid formed by
    z=x^2 +y^2 and z=2y-4x+1


    i think im supposed to do cylindrical coordinates.
    i got the limits for z and theta
    but not for r
    my limits were
    z: r^2 => 2rsin(theta) - 4rcos(theta) +1
    Theta: 0 => 2pi

    CAN U HELP ME PLZ? THANK YOU
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  2. #2
    Super Member
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    I think r is a function of \theta and the solution of:

    (x+1)^2+(y-1)^2=6

    when x=r\cos(\theta),\; y=r\sin(\theta)

    That's the domain in the x-y plane where the integration is being done (just complete the square when you equate both equations.

    Doing that I get (in cylindrical coordinates):

    \int_0^{2\pi}\int_0^{r(\theta)}\int_{g(r,\theta)}^  {h(r,\theta)} rdzdrd\theta

    where h(r,\theta)=r^2

    g(r,\theta)=2r\sin(\theta)-4r\cos(theta)+1

    and r(\theta) is one of the roots to the solution of that equation above. I don't know. Looks pretty messy to me. Maybe another easier way though.
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