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Math Help - need help on this integral~

  1. #1
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    need help on this integral~

    \int \int \int xy^2z^3 \,dx \,dy \,dz , where D is the solid in R3 bounded by the surface z = xy
    and the planes x = y, x = 1, and z = 0.

    I couldn't figure out the right limit.

    but I tried, the inner one is from z=0 to z=xy

    the middle one is from y=1 to y

    the outter one is from 0 to 1


    am i right?
    if not , can anyone explain to me?
    Last edited by yzc717; February 22nd 2009 at 06:27 PM.
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  2. #2
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    Quote Originally Posted by yzc717 View Post
    \int \int \int xy^2z^3 \,dx \,dy \,dz , where D is the solid in R3 bounded by the surface z = xy
    and the planes x = y, x = 1, and z = 0.

    I couldn't figure out the right limit.

    but I tried, the inner one is from z=0 to z=xy (*)

    the middle one is from y=1 to y (*)

    the outter one is from 0 to 1 (*)


    am i right?
    if not , can anyone explain to me?
    You have to be careful on the order of dx\, dy\, dz in your integral, typically we have

    \int_{P1}^{P_2} \int_{C_1}^{C_2} \int_{S_1}^{S_2} f(x,y,z)\,dV

    where S_1 \to S_2 surface to surface,
    where C_1 \to C_2 curve to curve and
    where P_1 \to P_2 point to point.

    The order of dx\,dy\,dz in dV gives the direction. So in

    \int \int \int xy^2z^3 \,dx \,dy \,dz the dx first is giving the surface to surface in the x direction. However, when giving your limits (in blue above) is suggests that what you're doing is the following

    \int_{x=0}^{x=1} \int_{y=0}^{y=x} \int_{z=0}^{z=xy} xy^2z^3 \,dz \,dy \,dx

    but we typically write

    \int_0^1 \int_0^x \int_0^{xy} xy^2z^3 \,dz \,dy \,dx
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  3. #3
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    Thank you , its now really make sense
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