Chapter 16.7, Problem 27E

Double integrals—transformation given To evaluate the following integrals, carry out these steps.

a.    Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.

b.    Find the limits of integration for the new integral with respect to u and v.

c.    Compute the Jacobian.

d.    Change variables and evaluate the new integral.

27.     ${\displaystyle \underset{R}{\iint }xydA}$. where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, –1); use x = u + v, y = u – v.