# Thread: Double integral change of variables.

1. ## 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
$

by making the substitution u = x, v = 2y-x.

Doing so I get:
$\int_0^2 \int_0^2 u^5ve^vdvdu
$

However...it should apparently become:
$\frac{1}{2}\int_0^2 \int_0^2 u^5ve^vdvdu
$

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.

2. 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.

$J(u,v ) = det\left[\begin{array}{cc}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{array}\right]$

3. Ok so basically I would have:
$
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?