Textbook Question
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 + √t, y = 2 - 4t; slope = -8
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 12.3.39
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside one leaf of r = cos 3θ
Verified step by step guidance1
First, understand the curve given: \(r = \cos 3\theta\) is a rose curve with 3 petals because the coefficient of \(\theta\) inside the cosine is 3. Each petal corresponds to a range of \(\theta\) where \(r\) is positive.
To find the area of one leaf (one petal), identify the interval of \(\theta\) that traces out exactly one petal. Since the rose has 3 petals evenly spaced over \(0\) to \(2\pi\), one petal corresponds to an interval of length \(\frac{2\pi}{3}\). Typically, one petal is traced from \(\theta = -\frac{\pi}{6}\) to \(\theta = \frac{\pi}{6}\) or from \(0\) to \(\frac{\pi}{3}\) depending on the petal chosen.
Recall the formula for the area enclosed by a polar curve between angles \(\alpha\) and \(\beta\) is:
\(\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\)
Substitute \(r = \cos 3\theta\) into the formula to get:
\(\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} (\cos 3\theta)^2 \, d\theta\)
Evaluate the integral by using the identity \(\cos^2 x = \frac{1 + \cos 2x}{2}\) to simplify the integrand before integrating. Then compute the definite integral over the chosen interval for one petal.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Graphing
Polar coordinates represent points using a radius and an angle, making it easier to describe curves like r = cos 3θ. Understanding how to sketch such curves helps visualize the region whose area is to be found, especially when the curve forms multiple 'leaves' or petals.
Recommended video:
05:32
Intro to Polar Coordinates
Area Calculation in Polar Coordinates
The area enclosed by a polar curve between two angles is found using the integral formula A = (1/2) ∫ (r(θ))^2 dθ. This formula accounts for the sector-like shape of regions in polar graphs, allowing precise calculation of areas bounded by curves like r = cos 3θ.
Recommended video:
05:32Intro to Polar Coordinates
Determining Limits of Integration for One Leaf
To find the area of one leaf of r = cos 3θ, it is essential to identify the correct interval of θ where the leaf exists. This involves solving r = 0 to find boundary angles and integrating over the range that traces exactly one petal without overlap.
Recommended video:
05:50One-Sided Limits
Related Practice
Textbook Question
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r = sin 3θ
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.
25y² - 4x² = 100
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Textbook Question
Multiple descriptions Which of the following parametric equations describe the same curve?
a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4
b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2
c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64
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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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Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = x²