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 ($\theta$)

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 $(r, \theta + 2\theta k)$ 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 $x = r \text{cos}(\theta)$ and $y = r \text{sin}(\theta)$ 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 $\tan(\theta) = \frac{y}{x}$ used to find $\theta$ from $(x, y)$, with careful attention to the correct quadrant.

Polar equation $r = f(\theta)$

A polar curve defined by specifying the radius $r$ as a function of the angle $\theta$, 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 $\theta$: $x(\theta) = r(\theta)\text{cos}(\theta)$ and $y(\theta) = r(\theta)\text{sin}(\theta)$.

Parametric slope rule (for polar curves)

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

$$r' (dr/d\theta)$$

The derivative of the polar radius function $r(\theta)$ with respect to $\theta$.

$\frac{dx}{d\theta}$ in polar form

For $x(\theta)=r(\theta)\text{cos}(\theta)$, the derivative is $\frac{dx}{d\theta} = r' \text{cos}(\theta) - r \text{sin}(\theta)$.

$\frac{dy}{d\theta}$ in polar form

For $y(\theta)=r(\theta)\text{sin}(\theta)$, the derivative is $\frac{dy}{d\theta} = r' \text{sin}(\theta) + r \text{cos}(\theta)$.

Polar slope formula

The formula $\frac{dy}{dx} = \frac{r' \text{sin}(\theta) + r \text{cos}(\theta)}{r' \text{cos}(\theta) - r \text{sin}(\theta)}$, 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 $d\theta$ has area $dA = \frac{1}{2}r^2 d\theta$ ($\theta$ in radians).

Polar area formula (single curve)

If $r=f(\theta)$ traces a region once from $\theta=a$ to $\theta=b$, the area is $A = \frac{1}{2}\frac{\text{d}}{d\theta} \bigg|_a^b (r(\theta))^2 d\theta$.

Polar area between two curves (washer-sector method)

For outer radius $r_{\text{outer}}$ and inner radius $r_{\text{inner}}$, area is $A = \frac{1}{2} \frac{\text{d}}{d\theta} \bigg|_a^b (r_{\text{outer}}(\theta)^2 - r_{\text{inner}}(\theta)^2) d\theta$, with bounds chosen to avoid double-tracing and to keep the correct outer/inner relationship.