Using the change $of$ variables $x=u^2-v^2,y=7uvto$ evaluate the integral ${}_R$ydA, where $Ris$ the region bounded $by$ the $x-$axis and the parabolas $y^2=7-7x$ and

$y^2=7+7x,y0.$

Solution

A similar region $Ris$ pictured $in$ the figure. $In$ Example $1,we$ discovered that $T(S)=R,$ where $Sis$ the square $[0,1][0,1].$ Indeed, the reason for making the change $of$

variables $to$ evaluate the integral $is$ that $Sis$ a much simpler region than $R.$ First $we$ need $to$ compute the Jacobian:

$\frac{\text{d}\text{e}\text{l}(x,y)}{\text{d}\text{e}\text{l}(u,v)}=|[\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{v}}$

$\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{v}}]|$

$=|[2u,-2v$

$7v,]|$

$=$

Therefore $by$ the theorem for the change $of$ variables $in$ a double integral