Textbook Question
41–44. Intersection points and area Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves
r = 3 sin θ and r = 3 cos θ
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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 12.2.49
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = x²
Verified step by step guidance1
Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(y = x^2\) to rewrite it in terms of \(r\) and \(\theta\).
This substitution gives \(r \sin{\theta} = (r \cos{\theta})^2\).
Simplify the right side to get \(r \sin{\theta} = r^2 \cos^2{\theta}\).
To isolate \(r\), divide both sides by \(r\) (assuming \(r \neq 0\)), resulting in \(\sin{\theta} = r \cos^2{\theta}\). Then solve for \(r\) to express the equation fully in polar form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cartesian and Polar Coordinate Systems
Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding the relationship between these systems is essential for converting equations between them.
Recommended video:
05:32
Intro to Polar Coordinates
Conversion Formulas Between Cartesian and Polar Coordinates
The key formulas for conversion are x = r cos(θ) and y = r sin(θ). To convert an equation from Cartesian to polar form, substitute x and y with these expressions and simplify to express the equation in terms of r and θ.
Recommended video:
05:32Intro to Polar Coordinates
Manipulating and Simplifying Trigonometric Expressions
After substitution, the resulting equation often involves trigonometric functions like sine and cosine. Being able to manipulate and simplify these expressions is crucial to rewrite the equation clearly in polar form, such as isolating r or expressing it as a function of θ.
Recommended video:
6:36Simplifying Trig Expressions
Related Practice
Textbook Question
102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.
Spiral of Archimedes: r = aθ
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Textbook Question
Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment.
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Textbook Question
80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.
r² - 8r cos(θ - π/2) = 9
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Textbook Question
Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.
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Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 3 csc θ
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