Mr. Integral
Math notes/Calculus II/Parametric and Polar Calculus

Polar Curves, Area and Length — Cheat sheet

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Core rule

x=rcos⁡θ,y=rsin⁡θ,A=12∫r2dθ,L=∫r2+r′2dθ.x=r\cos\theta,\quad y=r\sin\theta,\quad A=\frac12\int r^2d\theta,\quad L=\int\sqrt{r^2+r'^2}d\theta.

Watch for

Choose an interval tracing the intended loop once. Negative radii reverse direction; squaring does not prevent double-counting.