Textbook Question
Centroids
Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.5.27
Finding Lengths of Polar Curves
Find the lengths of the curves in Exercises 21–28.
The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4
Verified step by step guidance1
Recall the formula for the length of a curve given in polar coordinates: the arc length \( L \) from \( \theta = a \) to \( \theta = b \) is given by
\[
L = \int_a^b \sqrt{r(\theta)^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta
\]
Identify the given function and interval: \( r(\theta) = \cos^3\left(\frac{\theta}{3}\right) \) and \( 0 \leq \theta \leq \frac{\pi}{4} \).
Compute the derivative \( \frac{dr}{d\theta} \) using the chain rule:
First, write \( r(\theta) = \left( \cos\left(\frac{\theta}{3}\right) \right)^3 \).
Then,
\[
\frac{dr}{d\theta} = 3 \cos^2\left(\frac{\theta}{3}\right) \cdot \left(-\sin\left(\frac{\theta}{3}\right)\right) \cdot \frac{1}{3}
\]
Simplify this expression.
Substitute \( r(\theta) \) and \( \frac{dr}{d\theta} \) into the arc length formula:
\[
L = \int_0^{\frac{\pi}{4}} \sqrt{ \left( \cos^3\left(\frac{\theta}{3}\right) \right)^2 + \left( \frac{dr}{d\theta} \right)^2 } \, d\theta
\]
Set up the integral with the simplified expressions and prepare to evaluate it either analytically (if possible) or numerically to find the length of the curve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length of Polar Curves
The arc length of a curve defined in polar coordinates r(θ) from θ = a to θ = b is found using the formula L = ∫ from a to b √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for both the radial distance and the rate of change of r with respect to θ, providing the total length of the curve.
Recommended video:
06:29
Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, you need to compute the derivative dr/dθ of the given polar function r(θ). This involves using chain and power rules, especially when r is expressed as a composite function like cos³(θ/3). Accurate differentiation is essential for evaluating the integral correctly.
Recommended video:
05:32Intro to Polar Coordinates
Integration Techniques for Arc Length
After setting up the integral for arc length, you often need to simplify and evaluate it, which may require substitution or numerical methods if the integral is complex. Understanding how to handle trigonometric integrals and approximate definite integrals is crucial for finding the exact or approximate length.
Recommended video:
06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
Finding Cartesian from Parametric Equations
In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.
x = cos t, y = sin 3t
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Textbook Question
Identifying Graphs
Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x
Then find each parabola's focus and directrix.
Textbook Question
Ellipses and Eccentricity
Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.
Vertices: (±10,0)
Eccentricity: 0.24
Textbook Question
Surface Area
Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.
x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis
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Textbook Question
Tangent Lines to Parametrized Curves
In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.
x = sec² t − 1, y = tan t, t = −π/4
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