Compute the area of the region $D$ bounded by

$$xy = 1, \ xy = 9, \ \ \ xy^2 = 1, \ xy^2 = 36$$
in the first quadrant of the $xy$-plane.

(a) Graph the region $D$.

(b) Using the non-linear change of variables $u = xy$ and $v = xy^2$, find $x$ and $y$ as functions of $u$ and $v$.

$x = x(u,v) =$
$y = y(u,v) =$

(c) Find the determinant of the Jacobian for this change of variables.

$\displaystyle \left| \frac{\partial(x,y)}{\partial(u,v)} \right| = \mathrm{det}$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

=

(d) Using the change of variables, set up a double integral for calculating the area of the region $D$.

$\displaystyle \int\!\int_D dx \, dy = \int_a^b \int_c^d \left| \frac{\partial(x,y)}{\partial(u,v)} \right| du \, dv = \int$

$\displaystyle \int$

$du \, dv$

(e) Evaluate the double integral and compute the area of the region $D$.

Area =