Exercise 521Calculus › Multivariable calculus › Multiple integralsHigh schoolAdded 3 Nov 2026

A symmetric triangle integral

#symmetry#triangle#double integral

Statement

Compute $\displaystyle I=\iint_D (x+y)\,dx\,dy$ where $D$ is the triangle with vertices $(0,0),(2,0),(0,2)$ .

Hint

The triangle is symmetric in $x$ and $y$ : compute $\iint_D x$ only and multiply by $2$ .

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Setup. $D=\{(x,y):0\le x\le2,\ 0\le y\le2-x\}$ .

Compute $\iint_D x$ .
$\int_0^2\int_0^{2-x}x\,dy\,dx=\int_0^2 x(2-x)\,dx=\left[x^2-\frac{x^3}{3}\right]_0^2=4-\frac{8}{3}=\frac{4}{3}.$

Symmetry. $\iint_D y=\frac43$ , so $I=\frac83$ .

Conclusion. $I=8/3$ .