Chapter 26 — Key Takeaways
A structured recap of polar coordinates: the system itself, the gallery of curves, slope, the sector-derived area formula, area between curves, arc length, the common errors, and how it all connects forward and back. Section references point into index.md.
1. The Polar System and Conversion (§26.1)
A point is named $(r,\theta)$: $r$ is its distance from the pole (origin), $\theta$ the angle counterclockwise from the polar axis (positive $x$-axis).
| Direction | Formulas |
|---|---|
| Polar → Cartesian | $x = r\cos\theta$, $\quad y = r\sin\theta$ |
| Cartesian → polar | $r = \sqrt{x^2 + y^2}$, $\quad \theta = \texttt{atan2}(y,x)$ |
- Representation is not unique. $(r,\theta)$, $(r,\theta+2\pi)$ are the same point; the pole is $(0,\theta)$ for every $\theta$.
- Negative $r$ means "measure the angle, then walk backward": $(-r,\theta) = (r,\theta+\pi)$.
- Converting equations uses the "multiply by $r$" trick: create $r^2 = x^2+y^2$, $r\cos\theta = x$, $r\sin\theta = y$. (E.g. $r = 2\cos\theta \Rightarrow (x-1)^2+y^2 = 1$.)
2. The Curve Gallery (§26.2)
| Curve | Equation | Key feature |
|---|---|---|
| Circle (centered at pole) | $r = c$ | radius $c$ |
| Circle through pole | $r = 2a\cos\theta$ or $2a\sin\theta$ | radius $a$, center on an axis |
| Line through pole | $\theta = c$ | a fixed direction |
| Cardioid | $r = a(1+\cos\theta)$ | cusp at pole, reach $2a$ |
| Limaçon | $r = b + a\cos\theta$ | loop / dimple / convex by $|a|$ vs. $|b|$ |
| Rose | $r = a\cos(n\theta)$ | petal rule below |
| Archimedean spiral | $r = a\theta$ | equally spaced turns |
| Logarithmic spiral | $r = ae^{b\theta}$ | self-similar (equiangular) |
| Lemniscate | $r^2 = a^2\cos(2\theta)$ | figure-eight, two lobes |
| Conic (focus at pole) | $r = p/(1+e\cos\theta)$ | orbits — see §26.8 |
The petal rule. For the rose $r = a\cos(n\theta)$ ($n$ a positive integer): $n$ petals if $n$ is odd, $2n$ petals if $n$ is even. So $\cos(3\theta)$ has 3, but $\cos(4\theta)$ has 8. For even $n$ the negative-$r$ lobes draw a fresh set of petals in the gaps; for odd $n$ they retrace the existing ones.
3. Slope of a Polar Curve (§26.3)
Treat $r = f(\theta)$ as the parametric curve $x = f(\theta)\cos\theta$, $y = f(\theta)\sin\theta$ (Chapter 25). Then
$$\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}.$$
- Horizontal tangent: $dy/d\theta = 0$ (and $dx/d\theta \ne 0$).
- Vertical tangent: $dx/d\theta = 0$ (and $dy/d\theta \ne 0$).
- At the pole ($r = 0$ at $\theta = \theta_0$): the slope is $\tan\theta_0$, so the curve leaves the origin tangent to the ray $\theta = \theta_0$. Solving $f(\theta) = 0$ gives the directions a rose's petals or a cardioid's cusp leave the pole — no sketch needed.
4. Area: The Sector Formula (§26.5)
A polar region is sliced into thin circular sectors, not rectangles. A sector of radius $r$ and angle $d\theta$ has area $\tfrac12 r^2\,d\theta$ (it is the fraction $\tfrac{d\theta}{2\pi}$ of the disk $\pi r^2$). Summing and taking the limit:
$$\boxed{\,A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta\,}$$
Why $\tfrac12$ and $r^2$? They are the area of a pie slice. When you see $\tfrac12 r^2$, hear "sector." The radius enters twice — once for the slice's length, once for the width of its arc $r\,d\theta$ — which is why it is $r^2$, not $r$.
The deeper reason: this is the double integral $\int\!\!\int r\,dr\,d\theta$ done in the radial direction first, $\int_0^{f(\theta)} r\,dr = \tfrac12 f(\theta)^2$. The extra factor of $r$ in $dA = r\,dr\,d\theta$ is the Jacobian of the polar change of variables, justified in Chapter 33; polar double integrals over disks and sectors are Chapter 32.
5. Area Between Two Curves (§26.6)
With $r_\text{out} \ge r_\text{in} \ge 0$ on $[\alpha,\beta]$:
$$A = \frac{1}{2}\int_\alpha^\beta \big(r_\text{out}^2 - r_\text{in}^2\big)\,d\theta.$$
The polar version of "top minus bottom," but with squared radii (each piece is a sector). Procedure: find the intersections (and check the pole separately) to fix the limits, then integrate the difference of squares over each region where the ordering is constant.
6. Arc Length (§26.7)
From the parametric arc-length formula (Chapter 25), the cross terms cancel and $\sin^2 + \cos^2 = 1$ leaves:
$$\boxed{\,L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta\,}$$
The two terms are perpendicular displacements: $r^2$ is tangential motion (arc $r\,d\theta$ around the pole), $(r')^2$ is radial motion ($dr$ along the spoke). Pythagoras combines them into speed; arc length integrates speed.
7. Common Areas and Lengths (worth memorizing)
| Curve | Quantity | Value |
|---|---|---|
| Cardioid $r = 1+\cos\theta$ | area | $3\pi/2$ |
| Cardioid $r = 1+\cos\theta$ | arc length | $8$ |
| Circle $r = 2\cos\theta$ | area | $\pi$ |
| Rose $r = \cos(3\theta)$ | one petal | $\pi/12$ |
| Rose $r = \cos(2\theta)$ | one petal | $\pi/8$ |
| Lemniscate $r^2 = \cos(2\theta)$ | total (both lobes) | $1$ |
8. Application — Orbits (§26.8)
Under a central (gravitational) force, the orbit is a conic with the attractor at one focus:
$$r = \frac{p}{1 + e\cos\theta}, \qquad e = 0\ \text{circle},\ \ 0
For an ellipse: $p = a(1-e^2)$, $r_\text{min} = a(1-e)$ (perihelion), $r_\text{max} = a(1+e)$ (aphelion). Kepler's Second Law (equal areas in equal times) is the polar area element accumulating at a constant rate: $\tfrac{dA}{dt} = \tfrac12 r^2\dot\theta = \text{const}$ is exactly the conservation of angular momentum. The full geometry of these conics — foci, directrices, axes — is Chapter 27.
9. Common Pitfalls
- Dropping the $\tfrac12$ in area: it is $\tfrac12\int r^2\,d\theta$, never $\int r^2\,d\theta$.
- Quadrant errors in $\theta = \arctan(y/x)$: arctan only spans two quadrants. Check the signs of $x,y$ (or use
atan2). - Wrong limits of integration: integrate over the $\theta$-range that sweeps the region exactly once. For a single rose petal that is not $[0,2\pi]$; for an even-petaled rose, $[0,2\pi]$ double-counts.
- Forgetting negative $r$: $(-r,\theta) = (r,\theta+\pi)$; it draws real points (and real petals).
- The petal rule reversed: even $n$ gives $2n$ petals (not $n$); odd $n$ gives $n$.
- Intersections at the pole: two curves can cross where their equations never agree, because the pole has many names. Always check the pole separately and sketch.
10. Connections
- Chapter 25 (parametric): a polar curve is the parametric curve $x=f(\theta)\cos\theta$, $y=f(\theta)\sin\theta$; slope and arc length are inherited from there.
- Chapter 27 (conics): the full geometry of the ellipses, parabolas, and hyperbolas met here as $r = p/(1+e\cos\theta)$.
- Chapter 32 (double integrals) and Chapter 33 (change of variables / Jacobian): where the area element $r\,dr\,d\theta$ is justified rigorously and turned loose on integrating functions over polar regions.
- Chapter 28 and beyond: the "name a point by distance and direction" idea extends to cylindrical and spherical coordinates for 3-D problems with axial or spherical symmetry.
Skills Checklist
- [ ] Convert points and equations both directions, handling quadrants and negative $r$.
- [ ] Sketch every curve in the gallery; apply the petal rule from memory.
- [ ] Find $dy/dx$, including horizontal/vertical tangents and pole directions.
- [ ] Compute area with $\tfrac12\int r^2\,d\theta$ and choose correct limits.
- [ ] Compute area between curves after finding intersections.
- [ ] Compute arc length with $\int\sqrt{r^2+(r')^2}\,d\theta$.
- [ ] Read an orbit as $r = p/(1+e\cos\theta)$ and find perihelion/aphelion.
Reflection
The lesson outlasts any single formula: coordinates are a choice, and the right choice matches the symmetry of the problem. A planet's orbit is a one-line equation in $(r,\theta)$ and a tangle in $(x,y)$; a flower's petals, a galaxy's arms, an antenna's reach are all stories about angle. When the symmetry is rotational, polar wins — and the calculus (area from sectors, length from $\sqrt{r^2+(r')^2}$) meets you halfway. That habit of seeing the symmetry and picking the coordinate system starts here and runs through all of multivariable calculus.