AP Calculus BC glossary

Polar coordinates

Also called: Polar curve

Polar coordinates locate a point by its distance from the origin and the angle it makes with the positive horizontal axis. A polar curve expresses that distance as a function of the angle, which makes circles and roses far simpler to describe than in rectangular form.

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, \quad y = r\sin\theta

Polar curves are a special case of parametric equations with θ\theta as the parameter, which is why the slope formula is the parametric one applied to x=r(θ)cos⁡θx = r(\theta)\cos\theta and y=r(θ)sin⁡θy = r(\theta)\sin\theta.

Area in polar coordinates is not the usual integral of a height. Sectors, not rectangles, are the natural slice, giving A=12∫αβr2 dθA = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta.

The mistake

Forgetting the one half or the square in the polar area formula. Both come from the area of a circular sector and neither is optional.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)