Chapter 14, Problem 33RE

Let G be the solid in 3-space defined by the inequalities

$1-e^x\le y\le 3-e^x,1-y\le 2z\le 2-y,y\le e^x\le y+4$

(a) Using the coordinate transformation

$u=e^x+y,v=y+2z,w=e^x-y$ calculate the Jacobian $\partial \left(x,y,z\right)/\partial \left(u,v,w\right).$ Express your answer in terms of $u,v,\text{and}w.$

(b) Using a triple integral and the change of variables given in part (a), find the volume of G.