Exercise 1076Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Polar integral over a half-disk

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Statement

Compute $\displaystyle I=\iint_D\sqrt{x^2+y^2}\,dx\,dy$ where $D$ is the disk of radius $1$ centered at $(0,0)$ intersected with the half-plane $y\ge0$ .

Hint

Polar: $0\le r\le1$ , $0\le\theta\le\pi$ .

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Solution

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

Conclusion. $I=\pi/3$ .

Polar integral over a half-disk

Statement

Solution

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