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 limaçon r = 2 + cos θ
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.3.63
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete circle r = a sin θ, where a > 0
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = \alpha \) to \( \theta = \beta \):
\[
L = \int_{\alpha}^{\beta} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta
\]
Identify the given polar curve: \( r = a \sin \theta \), where \( a > 0 \). Since this represents a complete circle, the parameter \( \theta \) will vary from \( 0 \) to \( \pi \) to trace the entire circle.
Compute the derivative of \( r \) with respect to \( \theta \):
\[
\frac{d r}{d \theta} = a \cos \theta
\]
Substitute \( r \) and \( \frac{d r}{d \theta} \) into the arc length formula:
\[
L = \int_0^{\pi} \sqrt{(a \sin \theta)^2 + (a \cos \theta)^2} \, d\theta
\]
Simplify the expression inside the square root before integrating.
Evaluate the integral over the interval \( 0 \leq \theta \leq \pi \) to find the total length of the curve, which corresponds to the circumference of the circle described by \( r = a \sin \theta \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Curves
Polar coordinates represent points in the plane using a radius and an angle (r, θ). Polar curves are defined by equations relating r and θ, such as r = a sin θ, which describes a circle. Understanding how to interpret and plot these curves is essential for analyzing their properties.
Recommended video:
05:32
Intro to Polar Coordinates
Arc Length Formula for Polar Curves
The arc length of a polar curve r(θ) from θ = α to θ = β is given by the integral ∫ from α to β of √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length in polar form.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, one must compute the derivative dr/dθ of the polar function r(θ). This involves differentiating trigonometric functions like sin θ, which is crucial for evaluating the integral and finding the exact length of the curve.
Recommended video:
05:32Intro to Polar Coordinates
Related Practice
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 = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π
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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1
1
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Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
2 ≤ r ≤ 8
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Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
4x = -y²
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Textbook Question
31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.
x=2 sin 8t, y=2 cos 8t