Chapter 16.7, Problem 29E

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.

29.     ${\displaystyle \underset{R}{\iint }{x}^2\sqrt{x+2y}dA}$, where R = {(x, y): 0 ≤ x ≤ 2, –x/2 ≤ y ≤ 1 – x}; use x = 2u, y = v – u.