Exercise 1078Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Triple integral over a simplex

#triple integral#simplex#iterated

Statement

Compute $\displaystyle I=\iiint_D x\,dx\,dy\,dz$ where $D=\{x,y,z>0,\ x+y+z<1\}$ .

Hint

Integrate in the order $z,y,x$ ; bounds $0<z<1-x-y$ .

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Iterated.
$I=\int_0^1\int_0^{1-x}\int_0^{1-x-y}x\,dz\,dy\,dx=\int_0^1\int_0^{1-x}x(1-x-y)\,dy\,dx.$

Inner. $x\frac{(1-x)^2}{2}$ .

Outer. $I=\frac12\int_0^1 x(1-x)^2\,dx=\frac12\left(\frac12-\frac23+\frac14\right)=\frac1{24}$ .

Conclusion. $I=1/24$ .