$\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\= \|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\|$ 

$\int {\int }_{R}f\left(x\,y\right) \mathrm{dA}$ = $\int {\int }_{R\prime }f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right) \left|\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\right| \mathrm{dA}\prime$$$

Example 5.6.1

Let $R$ be the interior and boundary of the triangle whose vertices are $\left(1\,2\right)$, $\left(4\,9\right)$, and $\left(3\,5\right)$.

Example 5.6.2

Let $R$ be the interior and boundary of the parallelogram formed by the lines $x\+y\=0$, $x\+y\=2$, $2 y-3 x\=0$, $2 y-3 x\=7$.

Example 5.6.3

Let $R$ be the first-quadrant region bounded by the curves $C_1\:x^2-y^2\=1$, $C_2\:x^2-y^2\=4$, $C_3\:x^2\+y^2\=9$, $C_4\:x^2\+y^2\=16$.

Use the change of coordinates $u\=x^2-y^2\,v\=x^2\+y^2$ to evaluate the Cartesian integral $\int {\int }_{R}x y \mathrm{dA}$.$$

Example 5.6.4

Let $R$ be the region bounded by the curves $C_1\:x^2-y^2\=1$, $C_2\:x^2-y^2\=4$, $C_3\:2 x y\=1$, $C_4\:2 x y\=5$.

Express the Cartesian integral $\int {\int }_{R}f\left(x\,y\right) \mathrm{dA}$ in terms of the coordinates $u\=x^2-y^2\,v\=2 x y$.

Example 5.6.5

Let $R$ be the region bounded by the curves $C_1\:16x^{2}y\+16y^3\+8x-88 y\=0$, $C_2\:144x^{2}y\+144y^3-24 x-120 y\=0$, $C_3\:16x^{2}y\+16y^3-8 x-8 y\=0$.