Exercise 1794Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Triple integral with sine

#triple integral#sine#simplex

Statement

Compute $\displaystyle \iiint_V\frac{\sin x}{\pi-x-y}\,dx\,dy\,dz$ where $V=\{x,y,z\ge0,\ x+y+z\le\pi\}$ .

Hint

Integrate in $z$ first: the inner integral is $\sin x$ .

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Iterated.
$I=\int_0^\pi\int_0^{\pi-x}\int_0^{\pi-x-y}\frac{\sin x}{\pi-x-y}\,dz\,dy\,dx.$

Inner in $z$ . $z$ from $0$ to $\pi-x-y$ : the inner integral equals $\sin x$ .

In $y$ . $\int_0^{\pi-x}\sin x\,dy=(\pi-x)\sin x$ .

In $x$ . $I=\int_0^\pi(\pi-x)\sin x\,dx=\pi\int_0^\pi\sin x-\int_0^\pi x\sin x=\pi\cdot2-\pi=\pi$ .

Conclusion. $I=\pi$ .