Exercise 1784Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Linear change of variables

#change of variables#Jacobian#triangle

Statement

Compute $I=\iint_D(x-y)\,dx\,dy$ where $D$ is bounded by $x=0$ , $y=x+2$ , $y=-x$ , using the change of variables $u=x+y$ , $v=x-y$ .

Hint

Jacobian $|J|=1/2$ ; transform the bounds.

Show solutionHide solution

Transformed region. $D$ : $0\le u\le 2$ , $-2\le v\le -u$ .

Jacobian. $x=\frac{u+v}{2}$ , $y=\frac{u-v}{2}$ , so $|J|=\frac12$ .

Compute.
$I=\frac12\int_0^2\int_{-2}^{-u}v\,dv\,du=\frac12\int_0^2\left(\frac{u^2}{2}-2\right)du=\frac12\left(\frac43-4\right)=-\frac43.$

Conclusion. $I=-4/3$ .