Exercise 1075Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Double integral over a disk in polar coordinates

#polar coordinates#disk#Jacobian

Statement

Compute $\displaystyle I=\iint_D\frac{dx\,dy}{\sqrt{1+x^2+y^2}}$ where $D=\{x^2+y^2\le1\}$ .

Hint

Polar: $x=r\cos\theta$ , $y=r\sin\theta$ , $dx\,dy=r\,dr\,d\theta$ ; $0\le r\le1$ , $0\le\theta\le2\pi$ .

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Polar.
$I=\int_0^{2\pi}\int_0^1\frac{r}{\sqrt{1+r^2}}\,dr\,d\theta.$

Inner. $[\sqrt{1+r^2}]_0^1=\sqrt2-1$ .

Outer. $I=2\pi(\sqrt2-1)$ .

Conclusion. $I=2\pi(\sqrt2-1)$ .