Polar Curves (AP Calculus BC)

review Polar Coordinates

In the polar coordinate system, points are described using:

$$(r , \theta)$$

where:

• r = distance from the origin

• θ = angle from the positive x-axis

Conversion between systems:

Polar → Cartesian

$$x=r\cdot\cos(\theta) , y=r\cdot\sin(\theta)$$

Cartesian → Polar

$$r=\sqrt{x^{2}+y^{2}} , \theta=\tan^{-1}\left(\frac{y}{x}\right)$$

Polar Curves

A polar function is written as:

$$r=f(\theta)$$

As θ varies, the distance $r$ changes and traces a curve.

Example:

$$r=2\cos(\theta)$$

This is a circle with center (1,0) and radius $1$.

Common Polar Graphs (AP BC Important)

Circle

$$r=a\cos\theta or r=a\sin\theta$$

Rose Curve

$$r=a\cos(n\theta) or r=a\sin(n\theta)$$

• If n is odd: n petals

• If n is even: 2n petals

Cardioid (Heart-Shaped)

$$r=a(1+\cos\theta) or r=a(1+\sin\theta)$$

Lemniscate 

$$r^{2}=a^{2}\cos\theta or r^{2}=a^{2}\sin\theta$$

These appear frequently on AP BC exams.

Symmetry Tests (Useful for Sketching)

Symmetry about the x-axis:

Replace θ with −θ.

Symmetry about the y-axis:

Replace θ with π−θ.

Symmetry about the origin:

Replace $r$ with $r$.

Tangent Lines to Polar Curves (BC)

Given a polar curve r=f(θ):

Convert to parametric form:

$$x(\theta)=r\cos\theta , y(\theta)=t\sin\theta$$

The slope is given by:

$$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$$​​

derivatives respect by \theta:

$$\frac{dx}{d\theta}=r^\prime(\theta)\cos\theta-r\sin\theta$$

$$\frac{dy}{d\theta}=r^\prime(\theta)\sin\theta+r\cos\theta$$

Thus:

$$\frac{dx}{dy}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{r^\prime(\theta)\sin\theta+r\cos\theta}{r^\prime(\theta)\cos\theta-r\sin\theta}$$​​

Area in Polar Coordinates (BC)

The area swept out by a polar curve from θ=ato θ=b:

$$A=\frac{1}{2}\int{a}^{b}(r(\theta))^{2}d\theta$$​

Example:

Find the area inside one petal of:

$$r=2\sin(2\theta)$$

Step 1: Find bounds (one petal occurs between 0 to $$\frac{\pi}{2}$$​)
Step 2: Plug into area formula

$$A=\frac{1}{2}\int_{0}^{\frac{\pi}{2}}(2\sin(2\theta))^{2}d\theta$$

Area Between Two Polar Curves

If you have two curves:

$$r_{outer} , r_{inner}$$​

then:

$$A=\frac{1}{2}\int_{a}^{b}(r^{2}_{outer}-r^{2}_{inner})d\theta$$​