This is a very good conceptual problem for the topic of changing variables in 2D integrals. My problem in WebWork reads like this:

Find positive numbers $a$ and $b$ so that the change of variables $s=ax$ and $t=by$ transforms the integral $\int\int_R\,dx\,dy$ into

\[\int\int_T\left|\frac{\partial(x,y)}{\partial(s,t)}\right|\,ds\,dt\]

for the region $R$, the rectangle $0\le x\le 65$ and $0\le y\le 20$, and the region $T$ given by the square $0\le s,t\le 1$.

Hints: What’s incredibly nice about this problem is that we’re transforming one rectangular region $R$ to another rectangular region $T$ (and we are free to do this in an entirely linear fashion). So, if we know that $R$ is defined by $0\le x\le 65$ and $0\le  y \le 20$, then we simply squash our variables $x$ and $y$ by the appropriate factors (in this case $1/65$ and $1/20$) to achieve a square. So, we set $a=1/65$ and $b=1/20$. (The geometric interpretation of this transformation is something you definitely need to understand. Visit me in office hours if you don’t understand it.)

Given that $a=1/65$ and $b=1/20$, we have $x=65s$ and $y=20t$, so we know that

\[\left|\frac{\partial(x,y)}{\partial(s,t)}\right|=\left|\begin{array}{cc}65 & 0 \\ 0 & 20\end{array}\right|=1300.\]

Of course, we’re simply asking for the magnitude of the Jacobian here, and so we don’t consider including the $dx\,dy$, and the answer is simply 1300.

 

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