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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.78
Chapter 12, Problem 12.3.78

Spiral arc length Consider the spiral r=4θ, for θ≥0.


a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.

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Recall the formula for the length of a curve given in polar coordinates: \(L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta\).
Identify the given function: \(r(\theta) = 4\theta\), and the interval for \(\theta\) is from \(0\) to \(\sqrt{8}\).
Compute the derivative of \(r(\theta)\) with respect to \(\theta\): \(\frac{dr}{d\theta} = 4\).
Substitute \(r(\theta)\) and \(\frac{dr}{d\theta}\) into the arc length formula to get: \(L = \int_0^{\sqrt{8}} \sqrt{(4\theta)^2 + 4^2} \, d\theta = \int_0^{\sqrt{8}} \sqrt{16\theta^2 + 16} \, d\theta\).
Factor out the constant inside the square root and prepare for a trigonometric substitution: \(L = \int_0^{\sqrt{8}} 4 \sqrt{\theta^2 + 1} \, d\theta\). Use the substitution \(\theta = \sinh u\) or \(\theta = \tan u\) to simplify the integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Parametric Curves

In polar coordinates, a curve is described by r as a function of θ. Understanding how to express the curve in terms of r(θ) and θ is essential, as it allows conversion to parametric form for length calculations using x = r cos θ and y = r sin θ.
Recommended video:
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Intro to Polar Coordinates

Arc Length Formula for Polar Curves

The length of a curve defined in polar form r(θ) from θ = a to θ = b is given by the integral ∫ from a to b of √(r(θ)² + (dr/dθ)²) dθ. This formula combines the radius and its rate of change to measure the curve's length accurately.
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Arc Length of Parametric Curves

Trigonometric Substitution in Integration

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions. By substituting variables with trigonometric functions, the integral becomes easier to evaluate, which is useful when finding the arc length of curves like spirals.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=cos t+t sin t,y=sin t−t cos t; t=π/4

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 4 + sin θ; (4, 0) and (3, 3π/2)

Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


r = 3

Textbook Question

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.


Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

Textbook Question

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


(x - 1)² + y² = 1

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