Exercise 519Calculus › Multivariable calculus › Multiple integralsHigh schoolAdded 3 Nov 2026

Double integral over a triangular domain

#triangle#double integral#bounds

Statement

Compute $\displaystyle I=\iint_D \sin(x+y)\,dx\,dy$ where $D=\{x\ge0,\ y\ge0,\ x+y\le\pi\}$ .

Hint

For $x\in[0,\pi]$ , $y$ ranges from $0$ to $\pi-x$ .

Show solutionHide solution

Iterated. $I=\int_0^{\pi}\int_0^{\pi-x}\sin(x+y)\,dy\,dx$ .

Inner. $[-\cos(x+y)]_{y=0}^{y=\pi-x}=1+\cos x$ .

Outer. $I=\int_0^{\pi}(1+\cos x)\,dx=[x+\sin x]_0^{\pi}=\pi$ .

Conclusion. $I=\pi$ .