Exercise 1074Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Double integral over a disk tangent at origin

#polar coordinates#circle offset#integral

Statement

Compute $\displaystyle I=\iint_D\frac{dx\,dy}{\sqrt{x^2+y^2}}$ where $D=\{y\ge0,\ x^2+y^2-2x\le0\}$ .

Hint

The disk $(x-1)^2+y^2\le1$ has polar equation $0\le r\le2\cos\theta$ with $\theta\in[0,\pi/2]$ .

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Polar. $I=\int_0^{\pi/2}\int_0^{2\cos\theta}\frac{r}{r}\,dr\,d\theta=\int_0^{\pi/2}2\cos\theta\,d\theta=2$ .

Conclusion. $I=2$ .