Exercise 1212Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Line integral along a circle

#line integral#circle#parametrization

Statement

Compute $\displaystyle\oint_C(-y\,dx+x\,dy)$ where $C$ is the unit circle oriented counterclockwise.

Hint

$\gamma(t)=(\cos t,\sin t)$ , $t\in[0,2\pi]$ .

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Parametrization. $x=\cos t$ , $y=\sin t$ ; $dx=-\sin t\,dt$ , $dy=\cos t\,dt$ .

Integrand. $-y\,dx+x\,dy=\sin^2t\,dt+\cos^2t\,dt=dt$ .

Integral. $\oint_C = \int_0^{2\pi}dt=2\pi$ .

Conclusion. $\oint_C(-y\,dx+x\,dy)=2\pi$ (twice the area enclosed).