Chapter 16, 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.

73.     ${\displaystyle {\iint }_{}x{y}^{2}dA;R=\left\{\left(x,y\right):y/3\le x\le \left(y+6\right)/3,0\le y\le 3\right\}};\text{use}x=u+v/3,y=v.$