Textbook Question
Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 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 12.2.79
Circles in general Show that the polar equation
r² - 2r r₀ cos(θ - θ₀) = R² - r₀²
describes a circle of radius R whose center has polar coordinates (r₀, θ₀)
Verified step by step guidance1
Recall the relationship between polar and Cartesian coordinates: \(x = r \cos\theta\) and \(y = r \sin\theta\). Similarly, the center of the circle has Cartesian coordinates \(x_0 = r_0 \cos\theta_0\) and \(y_0 = r_0 \sin\theta_0\).
Rewrite the given polar equation \(r^2 - 2 r r_0 \cos(\theta - \theta_0) = R^2 - r_0^2\) by expressing \(\cos(\theta - \theta_0)\) using the cosine difference identity: \(\cos(\theta - \theta_0) = \cos\theta \cos\theta_0 + \sin\theta \sin\theta_0\).
Substitute the expressions for \(\cos\theta\) and \(\sin\theta\) in terms of \(x\) and \(y\), and similarly for \(\cos\theta_0\) and \(\sin\theta_0\), to rewrite the equation entirely in terms of \(x\), \(y\), \(x_0\), and \(y_0\).
Simplify the equation to get it into the standard form of a circle: \((x - x_0)^2 + (y - y_0)^2 = R^2\). This will involve expanding and rearranging terms to isolate the squared terms of \((x - x_0)\) and \((y - y_0)\).
Conclude that since the equation matches the standard form of a circle with center \((x_0, y_0)\) and radius \(R\), the original polar equation indeed describes a circle with center at polar coordinates \((r_0, \theta_0)\) and radius \(R\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Conversion to Cartesian Coordinates
Polar coordinates represent points using a radius and an angle (r, θ). To analyze curves like circles, it is often helpful to convert polar equations into Cartesian form using x = r cos θ and y = r sin θ. This conversion allows the use of familiar geometric interpretations and algebraic manipulations.
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Intro to Polar Coordinates
Equation of a Circle in Cartesian Coordinates
A circle with center (h, k) and radius R in Cartesian coordinates satisfies (x - h)² + (y - k)² = R². Recognizing this form after converting from polar coordinates confirms the geometric nature of the curve. Identifying the center and radius from the equation is key to understanding the shape described.
Recommended video:
05:32Intro to Polar Coordinates
Use of the Law of Cosines in Polar Form
The given polar equation resembles the law of cosines, which relates the lengths of sides in a triangle to the cosine of an included angle. Interpreting r, r₀, and R as sides and θ - θ₀ as the angle helps to understand the geometric meaning of the equation and its relation to a circle centered at (r₀, θ₀).
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3:37Convert Equations from Rectangular to Polar
Related Practice
Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 4 cos 2θ; at the tips of the leaves
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Textbook Question
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.
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Textbook Question
53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.
The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.
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Textbook Question
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x
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Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
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