Problem: Find positive numbers $a$ and $b$ such 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 elliptical region $x^2/25+y^2/36\le 1$ and the region $T$, the circle $s^2+t^2\le 1$.

Solution: This is very nearly an example that we did in class. It’s also taken from a webwork problem (16.7#4), which is taken from your test (16.7#11). Recall that the equation of an ellipse is given by

\[\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\]

where the $a$ and $b$ give intercepts on the $x$ and $y$-axes, respectively. What we want to do is bring those intercepts back to the unit circle in coordinates $s$ and $t$. Thus, we really need to set $s=ax$ where $a=1/5$ and $t=by$ where $b=1/6$. This gives us the function $x(s,t)=5s$ and $y(s,t)=6t$, which we need in order to find the Jacobian:

\[\left|\frac{\partial(x,y)}{\partial(s,t)}\right|=\left|\begin{array}{cc}5 & 0\\ 0 & 6\end{array}\right|=30.\]

Replacement Question: In a sentence, explain why one gets sunburned faster in Colorado.

Solution:¬†There is less atmosphere to block the sun’s rays.

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