Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
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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 12.2.50
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = 3
Verified step by step guidance1
Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Given the equation \(y = 3\), substitute \(y\) with its polar form: \(r \sin{\theta} = 3\).
Isolate \(r\) to express it in terms of \(\theta\): \(r = \frac{3}{\sin{\theta}}\).
Note the domain restrictions where \(\sin{\theta} \neq 0\) to avoid division by zero.
The polar form of the equation \(y = 3\) is therefore \(r = \frac{3}{\sin{\theta}}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates System
Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing curves that are difficult to express in Cartesian coordinates.
Recommended video:
05:32
Intro to Polar Coordinates
Conversion Formulas Between Cartesian and Polar Coordinates
To convert from Cartesian (x, y) to polar (r, θ), use r = √(x² + y²) and θ = arctan(y/x). Conversely, x = r cos(θ) and y = r sin(θ). These formulas allow rewriting equations from one coordinate system to the other.
Recommended video:
05:32Intro to Polar Coordinates
Expressing Cartesian Equations in Polar Form
To convert a Cartesian equation like y = 3 into polar form, substitute y with r sin(θ). This transforms the equation into r sin(θ) = 3, which can then be analyzed or manipulated using polar coordinates.
Recommended video:
3:47Introduction to Common Polar Equations
Related Practice
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.
10x² - 7y² = 140
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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 = cos t, y = sin² t; 0 ≤ t ≤ π
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Textbook Question
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
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