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Math Help - Double integral change of variables.

  1. #1
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    Double integral change of variables.

    I am slightly confused on the following problem:

    Change the order of integration of:
    \int_0^2 \int_{-x/2}^{x/2 + 1} x^5(2y-x)e^{2y-x}dydx<br />
    by making the substitution u = x, v = 2y-x.

    Doing so I get:
    \int_0^2 \int_0^2 u^5ve^vdvdu<br />

    However...it should apparently become:
    \frac{1}{2}\int_0^2 \int_0^2 u^5ve^vdvdu<br />

    That's where I am confused. I was able to see how to change the limits of integration, but I don't know where this 1/2 comes from. Consequently my answer is twice as much as it should be. Thanks for any help.
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  2. #2
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    You forgot about the Jacobian. It's the factor you have to multiply by when you do a change of variable in multiple integration. Like when you switch to polar coordinates and you have to multiply by r.

    J(u,v ) = det\left[\begin{array}{cc}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{array}\right]
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  3. #3
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    Ok so basically I would have:
    <br />
J(u,v ) = det\left[\begin{array}{cc}1&0\\-1&2\end{array}\right] = 2

    Because that is 2 I need to then multiply by 1/2 to compensate for it?
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