Exercise 1785Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Change of variables to simplify the integrand

#change of variables#Jacobian#triangular

Statement

Compute $\displaystyle J=\iiint_D\frac{z^3}{(y+z)(x+y+z)}\,dx\,dy\,dz$ where $D=\{x,y,z\ge0,\ x+y+z\le1\}$ using $u=x+y+z$ , $v=y+z$ , $w=z$ .

Hint

$\det J_h=1$ ; the region becomes $0\le w\le v\le u\le1$ .

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Transformation. $x=u-v$ , $y=v-w$ , $z=w$ ; $|J_h|=1$ .

Domain. $\Delta=\{0\le w\le v\le u\le1\}$ .

Integrand. $\frac{w^3}{uv}$ .

Compute.
$J=\int_0^1\int_0^u\int_0^v\frac{w^3}{uv}\,dw\,dv\,du=\int_0^1\int_0^u\frac{v^3}{4u}\,dv\,du=\int_0^1\frac{u^3}{16}\,du=\frac1{64}.$

Conclusion. $J=1/64$ .