Chapter 14.7, Problem 45ES

The formula obtained in part (b) of Exercise 43 is useful in integration problems where it is inconvenient or impossible to solve the transformation equations $u=f\left(x,y\right),\upsilon =g\left(x,y\right)$ explicitly for x and y in terms of $u\text{and}\upsilon .$ In these exercises, use the relationship $\frac{\partial \left(x,y\right)}{\partial \left(u,\upsilon \right)}=\frac{1}{\partial \left(u,\upsilon \right)/\partial \left(x,y\right)}$ to avoid solving for x and y in terms of $u\text{and}\upsilon .$

Use the transformation $u=x^2-y^2,\upsilon =x^2+y^2$ to find ${\displaystyle \underset{R}{\iint }xydA}$ where R is the region in the first quadrant that is enclosed by the hyperbolas $x^2-y^2=1,x^2-y^2=4$ and the circles $x^2+y^2=9,x^2+y^2=16.$