Chapter 16, Problem 61CRE

To determine

To calculate:

Sketch the region $D$bounded by the curves $y=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$x$}\right.,y=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2x$}\right.,y=2x,y=\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ in the first quadrant.

(a) The image of $D$under $F$

(b) Show that $\left|Jac\left(G\right)\right|=\frac{1}{2\left|v\right|}$

(c) Prove the formula by applying the change of variables formula

${\displaystyle \int {\displaystyle \underset{D}{\int }f}}\left(\frac{y}{x}\right)dxdy=\frac{3}{4}{\displaystyle \underset{1/2}{\overset{2}{\int }}\frac{f\left(v\right)dv}{v}}$

(d) Evaluate ${\displaystyle \int {\displaystyle \underset{D}{\int }\frac{y{e}^{\frac{y}{x}}}{x}dxdy}}$