Exercise 1072Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Cartesian vs polar computation

#polar vs cartesian#comparison#disk

Statement

Compare the difficulty of computing $\iint_{x^2+y^2\le1}(x^2+y^2)\,dx\,dy$ in Cartesian and polar coordinates.

Hint

Cartesian: use $y=\pm\sqrt{1-x^2}$ . Polar: integrand $r^2$ , Jacobian $r$ .

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Cartesian. $I=\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}(x^2+y^2)\,dy\,dx$ : requires trigonometric substitutions.

Polar. $I=\int_0^{2\pi}\int_0^1 r^3\,dr\,d\theta=2\pi\cdot\frac14=\frac{\pi}{2}$ .

Conclusion. Polar coordinates give an immediate result; Cartesian is much more involved.