Exercise 517Calculus › Multivariable calculus › Multiple integralsHigh schoolAdded 3 Nov 2026

Separable integrand on a square

#separability#product of integrals#square

Statement

Compute $\displaystyle I=\iint_{[0,1]^2}\frac{x^2}{1+y^2}\,dx\,dy$ .

Hint

Factor as a product of one-dimensional integrals.

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Factorisation.
$I=\left(\int_0^1 x^2\,dx\right)\left(\int_0^1\frac{dy}{1+y^2}\right)=\frac13\cdot\frac{\pi}{4}.$

Conclusion. $I=\pi/12$ .