Chapter 26 — Exercises

36 problems on polar coordinates: conversion, the curve gallery, slope, area, area between curves, arc length, and applications to orbital mechanics and engineering. Problems are tiered by difficulty. Answers to odd-numbered problems are in appendices/answers-to-selected.md.

Difficulty tiers: ⭐ routine (one step, direct formula) · ⭐⭐ standard (a short computation) · ⭐⭐⭐ multi-step (set up and evaluate) · ⭐⭐⭐⭐ challenge (modeling, proof, or synthesis)

Throughout, "convert" means give an exact answer; decimal approximations are for checking only. All angles are in radians unless stated otherwise.

Part A — Converting Points and Equations (⭐ / ⭐⭐)

Conversion is the bridge built in Section 26.1: $x = r\cos\theta$, $y = r\sin\theta$ one way; $r = \sqrt{x^2+y^2}$, $\theta = \texttt{atan2}(y,x)$ the other.

26.1 (⭐) Convert the polar point $(r,\theta) = (3, \pi/4)$ to Cartesian coordinates.

26.2 (⭐) Convert $(r,\theta) = (4, 2\pi/3)$ to Cartesian coordinates.

26.3 (⭐) Convert the polar point $(-2, \pi/6)$ to Cartesian coordinates. (Remember the negative-$r$ convention from Section 26.1.)

26.4 (⭐) Convert the Cartesian point $(1, -1)$ to polar coordinates with $r > 0$ and $0 \le \theta < 2\pi$.

26.5 (⭐) Convert the Cartesian point $(0, -3)$ to polar coordinates with $r>0$ and $0 \le \theta < 2\pi$. (The arctangent formula fails here — read the geometry.)

26.6 (⭐) Give three different polar names for the single point $(2, \pi/3)$, at least one using a negative $r$.

26.7 (⭐) The point $(-3, -3\sqrt{3})$ lies in which quadrant? Convert it to polar coordinates with $r>0$ and $0 \le \theta < 2\pi$, taking care with the quadrant.

26.8 (⭐⭐) Convert the Cartesian equation $x^2 + y^2 = 4y$ to polar form and simplify. Identify the curve.

26.9 (⭐⭐) Convert the polar equation $r = 2\cos\theta$ to Cartesian form by the "multiply by $r$" trick of Section 26.1. Identify the curve, including its center and radius.

26.10 (⭐⭐) Convert the vertical line $x = 5$ to polar form. Convert the horizontal line $y = -2$ to polar form.

26.11 (⭐⭐) Convert $r = \dfrac{4}{2\sin\theta - \cos\theta}$ to Cartesian form. What kind of curve is it?

Part B — The Curve Gallery and Petal Counting (⭐⭐)

These exercise the field guide of Section 26.2. For sketches, plot the curve over its natural $\theta$-range, mark where $r=0$, and label the maximum reach.

26.12 (⭐⭐) Sketch $r = 2\sin\theta$. Identify the curve, its center, and its radius.

26.13 (⭐⭐) Sketch the cardioid $r = 1 + \cos\theta$. Mark the cusp (where $r = 0$) and the point of maximum reach.

26.14 (⭐⭐) Sketch the limaçon $r = 1 + 2\cos\theta$. Does it have an inner loop, a dimple, or neither? Justify using the $|a|$ vs. $|b|$ criterion from Section 26.2.

26.15 (⭐⭐) Without plotting, state the number of petals of each rose and explain using the odd/even rule: (a) $r = \cos(3\theta)$, (b) $r = \cos(4\theta)$, (c) $r = \sin(5\theta)$, (d) $r = \sin(6\theta)$.

26.16 (⭐⭐) Sketch the three-petal rose $r = \cos(3\theta)$. Using the pole-tangent result of Section 26.3 ($r=0 \Rightarrow$ tangent along $\theta=\theta_0$), find the three directions in which the petals leave the origin.

26.17 (⭐⭐) Sketch the Archimedean spiral $r = \theta$ for $\theta \in [0, 4\pi]$. Roughly how far apart are successive turns along a fixed ray, and why is that spacing constant?

26.18 (⭐⭐) The lemniscate $r^2 = 4\cos(2\theta)$ exists only where $\cos(2\theta) \ge 0$. Find the two angular wedges on which it lives, and sketch the figure-eight.

Part C — Slope of a Polar Curve (⭐⭐⭐)

Use the slope formula of Section 26.3, $\dfrac{dy}{dx} = \dfrac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$, derived by treating $r=f(\theta)$ as a parametric curve (Chapter 25).

26.19 (⭐⭐⭐) For the cardioid $r = 1 + \cos\theta$, find all angles $\theta \in [0, 2\pi)$ where the tangent line is horizontal. (Worked partly in Section 26.3 — fill in the points.)

26.20 (⭐⭐⭐) For the same cardioid $r = 1 + \cos\theta$, find all angles where the tangent line is vertical (set $dx/d\theta = 0$, $dy/d\theta \ne 0$).

26.21 (⭐⭐⭐) Find the slope $dy/dx$ of the spiral $r = e^{\theta}$ at $\theta = 0$ and at $\theta = \pi/2$.

26.22 (⭐⭐⭐) Find the slope of the circle $r = 2\cos\theta$ at $\theta = \pi/3$. Convert the point to Cartesian and confirm the tangent direction makes geometric sense for a circle.

Part D — Area in Polar Coordinates (⭐⭐⭐)

The sector formula of Section 26.5: $A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$. The hardest part is choosing limits so the region is swept exactly once — sketch first.

26.23 (⭐⭐⭐) Find the area enclosed by the cardioid $r = 1 + \cos\theta$. (Confirm the value $3\pi/2$ derived in Section 26.5.)

26.24 (⭐⭐⭐) Find the area of one petal of the rose $r = \cos(3\theta)$, then the total area of all three petals. (Section 26.5 gives one petal $= \pi/12$.)

26.25 (⭐⭐⭐) Find the area of one petal of the four-petal rose $r = 2\sin(2\theta)$, then the total enclosed area.

26.26 (⭐⭐⭐) Find the area enclosed by the circle $r = 4\sin\theta$. Check against $\pi(\text{radius})^2$.

26.27 (⭐⭐⭐) Find the total area enclosed by the lemniscate $r^2 = 2\cos(2\theta)$ (both lobes).

Part E — Area Between Two Polar Curves (⭐⭐⭐)

Use Section 26.6: $A = \frac{1}{2}\int_\alpha^\beta (r_\text{out}^2 - r_\text{in}^2)\,d\theta$. Find intersections first to fix the limits, and check the pole separately (Section 26.1 warning).

26.28 (⭐⭐⭐) Find the area inside the cardioid $r = 1 + \cos\theta$ and outside the circle $r = 1$. (Confirm the value $2 + \pi/4$ from Section 26.6.)

26.29 (⭐⭐⭐) Find the area inside the circle $r = 3\sin\theta$ and inside the cardioid $r = 1 + \sin\theta$. (Find where the curves cross, then integrate the appropriate inner curve over each subinterval.)

26.30 (⭐⭐⭐) Find the area that lies inside both $r = 1$ and $r = 2\cos\theta$ (the lens-shaped overlap).

26.31 (⭐⭐⭐) Find the area inside one petal of $r = \cos(2\theta)$ but outside the circle $r = 1/2$.

Part F — Arc Length, Orbits, and Applications (⭐⭐⭐ / ⭐⭐⭐⭐)

Arc length uses Section 26.7: $L = \int_\alpha^\beta \sqrt{r^2 + (dr/d\theta)^2}\,d\theta$. Orbital and engineering problems draw on Section 26.8 ($r = p/(1+e\cos\theta)$, conic geometry deferred to Chapter 27).

26.32 (⭐⭐⭐⭐, arc length) Find the arc length of the cardioid $r = 1 + \cos\theta$ over one full trace, confirming $L = 8$ (Section 26.7). Then set up — but you need not evaluate — the arc length of one half of the cardioid and explain why symmetry lets you halve the work.

26.33 (⭐⭐⭐⭐, arc length / engineering) A spiral parking-garage ramp follows the Archimedean spiral $r = a\theta$. (a) Show that its arc length over $\theta \in [0,\Theta]$ is $$L = \frac{a}{2}\Big[\Theta\sqrt{1+\Theta^2} + \ln\big(\Theta + \sqrt{1+\Theta^2}\big)\Big].$$ (b) For $a = 2$ m and a ramp of two full turns ($\Theta = 4\pi$), compute $L$ to the nearest meter. (Real-World Application — Section 26.7: the contractor orders handrail by this number.)

26.34 (⭐⭐⭐⭐, astronomy / orbital mechanics) Halley's Comet orbits the Sun on an ellipse with semi-major axis $a \approx 17.8$ AU and eccentricity $e \approx 0.967$. (a) Using $r_\text{min} = a(1-e)$ and $r_\text{max} = a(1+e)$ from Section 26.8, find the comet's perihelion and aphelion distances in AU. (b) Write the comet's orbit in polar form $r = p/(1 + e\cos\theta)$ by computing $p = a(1-e^2)$. (c) Kepler's Second Law (Section 26.8) says the comet sweeps equal areas in equal times. Explain in one or two sentences why this forces the comet to move far faster at perihelion than at aphelion.

26.35 (⭐⭐⭐⭐, electrical engineering) A dipole antenna has the planar radiation pattern $G(\theta) = |\cos\theta|$ (Section 26.8, antenna application). (a) Sketch the pattern in polar form and identify the figure-eight shape. (b) The "total radiated power" in this 2-D idealization is modeled by $P = \frac{1}{2}\int_0^{2\pi} G(\theta)^2\,d\theta$ — the polar-area machinery of Section 26.5 applied to power. Evaluate it. (c) The half-power beamwidth is the angular width where $G(\theta) \ge G_\text{max}/\sqrt{2}$. Find it for this dipole.

26.36 (⭐⭐⭐⭐, data science) Wind direction at a weather station is recorded as a rose histogram: over a month, the relative frequency of wind blowing from angle $\theta$ is modeled by $r = f(\theta) = 1 + \tfrac12\cos(2\theta)$ (a mild bias toward the east–west axis). (a) Using the polar-area "total mass" idea from the chapter's Modeling Portfolio prompt (Section 26.9), compute $\frac{1}{2}\int_0^{2\pi} f(\theta)^2\,d\theta$. (b) Interpret what a larger value of this integral would tell you about the wind distribution's overall spread.

Hints and Selected Strategies

• 26.8, 26.9: Multiply through by $r$ to create $r^2 = x^2+y^2$ and $r\cos\theta = x$, $r\sin\theta = y$, then complete the square.
• 26.19–26.20: Horizontal tangents need $dy/d\theta = 0$; vertical tangents need $dx/d\theta = 0$. The cusp ($r=0$) is special — check it separately.
• 26.24–26.25: A single petal is swept where $r \ge 0$ between consecutive zeros of $r$. For $r = \cos(3\theta)$ that is $-\pi/6 \le \theta \le \pi/6$.
• 26.27: By symmetry, total lemniscate area $= 2 \times \frac{1}{2}\int_{-\pi/4}^{\pi/4} 2\cos(2\theta)\,d\theta$. Expect the clean answer $2$.
• 26.29: The curves $r=3\sin\theta$ and $r = 1+\sin\theta$ meet where $3\sin\theta = 1+\sin\theta$, i.e. $\sin\theta = \tfrac12$. The boundary switches from one curve to the other at those angles.
• 26.34: Halley's huge eccentricity ($e \approx 0.967$) is why it spends most of a 76-year period far out and only weeks near the Sun — a direct consequence of equal areas in equal times.

Total: 36 exercises (7 ⭐, 11 ⭐⭐, 13 ⭐⭐⭐, 5 ⭐⭐⭐⭐). Applied fields represented: astronomy (orbital mechanics), civil/mechanical engineering (spiral ramp), electrical engineering (antenna patterns), and data science (wind rose).