Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π
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 12.2.20
15–22. Sets in polar coordinates Sketch the following sets of points.
1 < r < 2 and π/6 ≤ θ ≤ π/3
Verified step by step guidance1
Understand the problem: We need to sketch the set of points in polar coordinates where the radius \(r\) satisfies \(1 < r < 2\) and the angle \(\theta\) satisfies \(\frac{\pi}{6} \leq \theta \leq \frac{\pi}{3}\).
Recall that in polar coordinates, each point is represented by \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis.
Identify the region for \(r\): Since \(1 < r < 2\), the points lie between two circles centered at the origin with radii 1 and 2. This forms an annular (ring-shaped) region between these two circles.
Identify the region for \(\theta\): The angle \(\theta\) ranges from \(\frac{\pi}{6}\) to \(\frac{\pi}{3}\), which corresponds to a sector of the annulus between these two rays starting from the origin at these angles.
To sketch, draw two concentric circles with radii 1 and 2, then shade the region between them. Next, draw two rays from the origin at angles \(\frac{\pi}{6}\) and \(\frac{\pi}{3}\), and shade the sector of the annulus bounded by these rays.
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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 a plane using a radius and an angle, denoted as (r, θ). The radius r measures the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for interpreting and sketching regions defined by inequalities in r and θ.
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Intro to Polar Coordinates
Inequalities in Polar Coordinates
Inequalities like 1 < r < 2 and π/6 ≤ θ ≤ π/3 define a region in the polar plane. The radius bounds specify a ring-shaped area between two circles, while the angle bounds restrict the sector of the circle. Combining these inequalities helps visualize the exact portion of the plane to sketch.
Recommended video:
05:32Intro to Polar Coordinates
Graphing Regions in Polar Coordinates
Sketching sets in polar coordinates involves plotting points that satisfy given radius and angle constraints. This typically results in sectors or annular sectors. Recognizing how to translate these inequalities into geometric shapes aids in accurately drawing the specified region.
Recommended video:
05:32Intro to Polar Coordinates
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.
4x² - y² = 16
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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 = 1 + sin θ and r = 1 + cos θ
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Textbook Question
69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
x² = -6y; (-6, -6)
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