Exercise 1071Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Boundary effects

#boundary#integration bounds#evaluation

Statement

Compute $\displaystyle\iint_D xy\,dx\,dy$ where $D=\{(x,y):0\le x\le y\le1\}$ .

Hint

Triangle: $0\le x\le y\le1$ , so for $y\in[0,1]$ , $x$ runs from $0$ to $y$ ; or for $x\in[0,1]$ , $y$ runs from $x$ to $1$ .

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Iterated. $\int_0^1\int_x^1 xy\,dy\,dx=\int_0^1 x\frac{1-x^2}{2}\,dx=\frac{1}{2}\left(\frac12-\frac14\right)=\frac18$ .

Conclusion. $I=1/8$ .