Module 32: Polar coordinates, polar calculus, and conics

A. Prerequisites and observable goals#

B. The central idea: locate a point by direction and signed distance#

Polar coordinates represent a point by $(r,\theta)$ with $x=r\cos\theta$, $y=r\sin\theta$. A negative $r$ places the point in the direction opposite $\theta$. The representation is not unique: $(r,\theta)$ and $(r,\theta+2\pi)$ agree, and $(-r,\theta+\pi)$ agrees as well. At the origin every angle represents the same point. These redundancies make tracing intervals important in integration.

C. Calculus formulas and their geometric conditions#

For $r=r(\theta)$, differentiation of its Cartesian parameterization gives

when the denominator is nonzero. Arc length on a smooth tracing is

A polar sector bounded by a radial graph and rays, with the intended region swept once, has area

For two nonnegative radial boundaries on the same rays, use $\tfrac12\int(R^2-r^2)\dd\theta$. Before using this for intersections or negative radii, confirm that the angular interval really describes the intended region without duplicate tracing. Polar calculus: [O2Polar].

A conic with a focus at the origin may have form $r=\ell/(1\pm e\cos\theta)$ or $\ell/(1\pm e\sin\theta)$, with $\ell>0$ and eccentricity $e\ge0$. For nondegenerate standard branches, $0<e<1$ gives an ellipse, $e=1$ a parabola, and $e>1$ a hyperbola; $e=0$ gives a circle. Here $\ell$ is a scale parameter, not generally the semimajor axis. The denominator’s zeros and signs restrict the branch represented.

D. Why polar area has a half and a square#

A circular sector of radius $r$ and angle $\Delta\theta$ radians has area $r^2\Delta\theta/2$. As the angular partition becomes fine, varying-radius sectors give the integral. This is not circumference times width: area accumulates entire radial wedges. The factor $1/2$ and radians both arise from the sector formula.

For length, squaring the Cartesian derivatives gives $(r'\cos\theta-r\sin\theta)^2+(r'\sin\theta+r\cos\theta)^2=(r')^2+r^2$; the cross terms cancel. This derives the formula directly from the parametric arc-length rule rather than guessing it from the area expression.

E. A tracing-first procedure#

Sketch key angles, zeros of $r$, maximum distances, and any negative-radius intervals. Convert to Cartesian form when that simplifies a circle or conic. For area, find where the intended loop begins and ends and whether it is traced once. For intersections, remember that equal $r$ at equal angles is useful but does not by itself detect every shared point represented with different polar coordinates; check the origin and the geometry. For slopes, evaluate both $r$ and $r'$ at the angle before simplifying.

F. Worked examples#

A circle disguised as a polar equation. For $r=4\cos\theta$, multiply by $r$: $r^2=4r\cos\theta$ becomes $x^2+y^2=4x$, or $(x-2)^2+y^2=4$. The interval $-\pi/2\le\theta\le\pi/2$ has $r\ge0$ and traces the circle once. Integrating area over $0\le\theta\le2\pi$ would count the disk twice.

A cardioid area. For $r=1+\cos\theta$ on $[0,2\pi]$, the curve is traced once with nonnegative radius. Its area is

The negative-valued cosine does not make the radius negative here; the added one matters.

G. Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

H. Complete solutions: Module 32#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

I. Retention and readiness#

Retrieve M31.05, M18.08, and M04.01. Explain why a full $2\pi$ interval is not always one tracing. Complete the Calculus II assessment next, recording separately mistakes in method choice, endpoint conditions, and computation.