I've used Jacobians before in multivariable calculus to simplify integrals, but I'm lost when I need to find the substitutions myself. Today on the quiz, there was the problem
$\int\int_{R} xy dxdy$ given a region $R$ bounded by
$y=\frac{2x+5}{3}$
$y=\frac{2x+2}{3}$
$y=\frac{-5-5x}{2}$
$y=\frac{-3-5x}{2}$
I know how to calculate the Jacobian if I know the substitutions, but I'm lost on how to calculate the substitutions. I've tried the following:
$u = xy, v=1$
I get a feeling, that given the limits, that both $u$ and $v$ will be linearly related to $x$ and $y$.
Rewrite your bounding equations to solve for constant factors like so:
$$2x - 3y = -5 \\2x - 3y = -2 \\ 5x + 2y = -5 \\ 5x + 2y = -3$$
One way to look at this is that you're solving for constants (on the left), so to speak.
How can we think about this in general terms? Well, imagine a rectangular region on a Cartesian coordinate system, from $(0,0)$ to $(a,0)$ to $(a,b)$ to $(0,b)$. This is bounded by four curves: $x = 0$, $x=a$, $y=0$, and $y=b$. The region is bounded by four curves of constant coordinates, two for each coordinate.
I think you should now be able to see here how you can define the new coordinates so that the region's boundaries can be expressed just as simply.