Polar curves, tangents and areas

What you’ll learn

• Convert between polar and Cartesian descriptions.
• Calculate polar areas and locate horizontal or vertical tangents.

Understand the idea

Polar coordinates describe a point by its distance $r$ from the pole and angle $\theta$ from the initial line. In this unit take $r\ge0$, so sketch only angle intervals where the radius is non-negative. Symmetry and values at key angles help reveal the curve.

A polar area is built from thin sectors, giving one half of the integral of $r^2$. Choose bounds that trace the intended region exactly once. For tangents, differentiate $x=r\cos\theta$ and $y=r\sin\theta$; a horizontal tangent normally has $dy/d\theta=0$ with $dx/d\theta\ne0$, and conversely for a vertical tangent.

Key formulae

$$x=r\cos\theta,\qquad y=r\sin\theta$$

$$A=\frac12\int_\alpha^\beta r^2\,d\theta$$

$$\frac{dy}{dx}=\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$$

Worked example

Find the area enclosed by $r=4\cos\theta$, using $-\pi/2\le\theta\le\pi/2$.

1. The chosen interval keeps $r\ge0$ and traces the circle once.
2. The area is $8\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta$.
3. Use $\cos^2\theta=(1+\cos2\theta)/2$; the integral is $\pi/2$.

Answer: $4\pi$ square units.