Textbook Question
Lengths of Curves
Find the lengths of the curves in Exercises 25–30.
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 11.3.36
Polar to Cartesian Equations
Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.
r² = 4r sin θ
Verified step by step guidance1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r^2 = 4r \sin \theta\).
Divide both sides of the equation by \(r\) (noting that \(r \neq 0\)) to simplify: \(r = 4 \sin \theta\).
Substitute \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\) into the simplified equation: \(\sqrt{x^2 + y^2} = 4 \cdot \frac{y}{\sqrt{x^2 + y^2}}\).
Multiply both sides by \(\sqrt{x^2 + y^2}\) to eliminate the denominator, then simplify and rearrange the resulting equation to express it purely in terms of \(x\) and \(y\). This will give the Cartesian form of the curve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar to Cartesian Coordinate Conversion
Polar coordinates (r, θ) relate to Cartesian coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Understanding these relationships allows one to rewrite polar equations in terms of x and y, facilitating analysis using Cartesian methods.
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05:32
Intro to Polar Coordinates
Algebraic Manipulation of Equations
After substituting polar expressions with Cartesian equivalents, algebraic skills are essential to simplify and rearrange the equation. This process helps in identifying the standard form of the curve and makes it easier to classify the graph.
Recommended video:
Guided course
08:02Parameterizing Equations
Graph Identification from Equations
Recognizing the type of graph from its Cartesian equation involves knowledge of common curves like circles, lines, parabolas, and ellipses. Identifying these shapes helps in visualizing and describing the graph represented by the original polar equation.
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04:47Introduction to Parametric Equations
Related Practice
Textbook Question
Examples of Polar Equations
[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.
r = −2 cos θ
Textbook Question
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
x - y = 3
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Textbook Question
Finding Parametric Equations
In Exercises 31–36, find a parametrization for the curve.
the ray (half line) with initial point (-1,2) that passes through the point (0,0)
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Textbook Question
Finding Parametric Equations
In Exercises 31–36, find a parametrization for the curve.
the line segment with endpoints (-1,3) and (3,-2)
Textbook Question
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π