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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.3.78
Capitolo 12, Problema 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.

Guida verificata passo dopo passo
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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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.
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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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Percorso guidato
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Introduction to Trigonometric Functions
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