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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 12.3.55
Capitolo 12, Problema 12.3.55

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


The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

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First, understand the problem: we need to find the area of the region that lies inside the outer loop but outside the inner loop of the limaçon given by the polar equation \(r = 3 - 6 \sin \theta\).
Identify the points where the limaçon intersects itself or where the loops begin and end by solving \(r = 0\), i.e., solve \(3 - 6 \sin \theta = 0\) for \(\theta\). This will give the boundary angles separating the inner and outer loops.
Recall the formula for the area enclosed by a polar curve between two angles \(\alpha\) and \(\beta\) is \(\frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). We will use this to find the areas of the outer and inner loops separately.
Set up the integral for the area of the outer loop by integrating \(\frac{1}{2} (3 - 6 \sin \theta)^2\) over the interval corresponding to the outer loop (determined from step 2). Similarly, set up the integral for the inner loop over its corresponding interval.
Subtract the area of the inner loop from the area of the outer loop to find the area of the region inside the outer loop but outside the inner loop.

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Polar Coordinates and Graphing Polar Curves

Polar coordinates represent points using a radius and an angle, making it easier to describe curves like limaçons. Understanding how to plot and interpret the graph of r = 3 - 6 sin θ is essential to visualize the inner and outer loops and identify the regions described.
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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 of (1/2)r² dθ. Calculating areas of regions bounded by loops requires setting up integrals with correct limits and sometimes subtracting areas to find regions inside one loop but outside another.
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05:32
Intro to Polar Coordinates

Identifying and Distinguishing Inner and Outer Loops of a Limaçon

Limaçons can have inner loops when the curve crosses the pole multiple times. Recognizing the angles where r = 0 helps determine the boundaries of inner and outer loops, which is crucial for setting integration limits to find the area between these loops.
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2:52
Limacons Example 2
Pratica correlata
Domanda del libro di testo

81–88. Arc length Find the arc length of the following curves on the given interval.


x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

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Domanda del libro di testo

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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Domanda del libro di testo

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

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Domanda del libro di testo

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 


r = 3/(1 - cos θ)

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Domanda del libro di testo

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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Domanda del libro di testo

63–74. Arc length of polar curves Find the length of the following polar curves.


The complete circle r = a sin θ, where a > 0

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