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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.55
Chapter 12, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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 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.
Recommended video:
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.
Recommended video:
2:52
Limacons Example 2
Related Practice
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.


A circle centered at the origin with radius 4, generated counterclockwise

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Textbook Question

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)

Textbook Question

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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Textbook Question

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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Textbook Question

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

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 = 1 - sin θ; (1/2, π/6)

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