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Math Help - Triple integral problem

  1. #1
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    Triple integral problem

    I need to set the limits of integration for :


    The volume of the region between z = x and the surface z = x^2 and the planes y = 0 and y = 3


    I am having trouble figuring these bounds!! I know that y goes from 0 to 3. Would z be bound from x to x^2 and x from square root of z to z?????

    Any help would much appreciated. Frostking
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  2. #2
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    If you have a volume bounded by g(x,y)\leq z \leq f(x,y) then you can calculate the volume as \iint_D (f(x,y)-g(x,y))dxdy where D is the projection on the xy-plane.

    We already know the limits for y, and z=x=x^2 \Longleftrightarrow x=0,\ x=1
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  3. #3
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    Quote Originally Posted by Frostking View Post
    I need to set the limits of integration for :


    The volume of the region between z = x and the surface z = x^2 and the planes y = 0 and y = 3


    I am having trouble figuring these bounds!! I know that y goes from 0 to 3. Would z be bound from x to x^2 and x from square root of z to z?????

    Any help would much appreciated. Frostking
    setting the equations for z equal we get

    x=x^2 \iff x^2-x=0 \iff x(x-1)=0 so x =0 \mbox{ or } x=1


    \int_{0}^{1} \int_{0}^{3} \int_{x^2}^{x}f(x,y,z)dzdydx

    Since z only depends on x we could switch the order of the x and y integration becuase they don't depend on each other

    \int_{0}^{3} \int_{0}^{1} \int_{x^2}^{x}f(x,y,z)dzdxdy
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