AP Calculus BC Unit 9 Notes: Polar Curves, Slopes, and Areas

Polar coordinates

A coordinate system that locates a point by its directed distance from the origin (r) and an angle from the positive x-axis (θ), written (r, θ).

Pole

The origin in the polar coordinate system; the reference point from which r is measured.

Polar radius (r)

The directed distance from the pole to the point; it can be negative, which indicates the point lies in the opposite direction.

Polar angle (θ)

The angle (typically measured from the positive x-axis) that determines the direction to the point in polar coordinates.

Angle periodicity in polar form

The fact that the same point can be written as (r, θ + 2πk) for any integer k, because angles repeat every full rotation.

Negative-radius equivalence

The identity (r, θ) = (−r, θ + π), meaning a negative radius flips the direction by π radians.

Polar-to-Cartesian conversion

The relationships x = r cosθ and y = r sinθ connecting polar and Cartesian coordinates.

Cartesian-to-polar distance formula

The relationship r² = x² + y², derived from the Pythagorean theorem.

Tangent-angle relationship (quadrant-aware)

The relationship tanθ = y/x used to find θ from (x, y), with careful attention to the correct quadrant.

Polar equation r = f(θ)

A polar curve defined by specifying the radius r as a function of the angle θ, often visualized as a “radar sweep.”

Graph of r = c (constant)

A circle centered at the origin with radius |c|.

Polar symmetry about the x-axis test

If replacing θ with −θ leaves the equation unchanged, the graph is symmetric about the x-axis.

Polar symmetry about the y-axis test

If replacing θ with π − θ leaves the equation unchanged, the graph is symmetric about the y-axis.

Polar symmetry about the origin test

If replacing θ with θ + π leaves the equation unchanged, the graph is symmetric about the origin.

Polar curve as a parametric curve

Treating x and y as functions of θ: x(θ) = r(θ)cosθ and y(θ) = r(θ)sinθ.

Parametric slope rule (for polar curves)

The tangent slope is dy/dx = (dy/dθ)/(dx/dθ) when θ is the parameter.

r′ (dr/dθ)

The derivative of the polar radius function r(θ) with respect to θ.

dx/dθ in polar form

For x(θ)=r(θ)cosθ, the derivative is dx/dθ = r′cosθ − r sinθ.

dy/dθ in polar form

For y(θ)=r(θ)sinθ, the derivative is dy/dθ = r′sinθ + r cosθ.

Polar slope formula

The formula dy/dx = (r′sinθ + r cosθ)/(r′cosθ − r sinθ), obtained from the parametric slope rule.

Horizontal tangent condition (polar/parametric)

A horizontal tangent occurs when dy/dθ = 0 and dx/dθ ≠ 0 at the same θ.

Vertical tangent condition (polar/parametric)

A vertical tangent occurs when dx/dθ = 0 and dy/dθ ≠ 0 at the same θ.

Polar sector area (infinitesimal)

A thin sector of radius r and angle dθ has area dA = (1/2)r² dθ (θ in radians).

Polar area formula (single curve)

If r=f(θ) traces a region once from θ=a to θ=b, the area is A = (1/2)∫_a^b (r(θ))² dθ.

Polar area between two curves (washer-sector method)

For outer radius router and inner radius rinner, area is A = (1/2)∫a^b (router(θ)² − r_inner(θ)²) dθ, with bounds chosen to avoid double-tracing and to keep the correct outer/inner relationship.