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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.

Verified step by step guidance
1
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.
Recommended video:
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 1/(2 - 2 sin θ)

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Textbook Question

What is the polar equation of the horizontal line y = 5?

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Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = r − 1, y = r³; −4 ≤ r ≤ 4

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Textbook Question

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the curve r = √(cos θ)

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Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The left half of the parabola y=x ² +1, originating at (0, 1)

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