I'm working on a question that asks to:
Find the area in the first quadrant bounded by the curves; $\ xy = 1, xy=5, y=e^2x, y=e^5x $.
I would very much appreciate help solving this question (including the method of how to find the transformation expressions for $\ u$ and $v$ to use in the Jacobian). Thanks!
On
An idea:
Try to write everything as function of $\;x\;$ to find intersection points. After all we have two hyperbolas and two straight lines:
$$\begin{cases}y=\frac1x\\{}\\y=\frac5x\\{}\\y=e^2x\\{}\\y=e^5x\end{cases}\;\;\implies\begin{cases}e^2x=\frac1x\implies x=\frac1e\\{}\\e^2x=\frac5x\implies x=\frac{\sqrt5}e\end{cases}$$
and do the same with $\;e^5x\;$ . Now, since $\;\frac5x>\frac1x\;$ and $\;e^2x<e^5x\;\;\forall\,x>0$ , find out the limits...and you may need more than one integral as the domain of integration isn't a simple one (a geometric sketch could help a lot)
A change of variable $u=x\,y$, $v=y/x$ will transform the domain of integration into a rectangle.