Chapter 13, Problem 75RE

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

a.    Sketch the original region of integration R and the new region S 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.    Chance variables and evaluate the new integral.

75.     ${\displaystyle \underset{}{\iint }{x}^2\sqrt{x+2y}dA;R=\left\{\left(x,y\right):0\le x\le 2,-x/2\le y\le 1-x\right\}}$; use x = 2u, y = v – u.