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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.20
Capitolo 12, Problema 12.2.20

15–22. Sets in polar coordinates Sketch the following sets of points.


1 < r < 2 and π/6 ≤ θ ≤ π/3

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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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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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05:32
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.
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05:32
Intro 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.
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05:32
Intro to Polar Coordinates
Pratica correlata
Domanda del libro di testo

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 lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

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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

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45–60. Areas of regions Find the area of the following regions.


The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=tan t, y=sec ² t−1 

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25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

r = 4 cos θ

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