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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.R.49
Chapter 12, Problem 12.R.49

44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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First, understand the problem: we need to find the area of the region that lies inside the cardioid given by \(r = 1 + \cos\theta\) and outside the cardioid given by \(r = 1 - \cos\theta\).
Determine the points of intersection of the two cardioids by setting their equations equal: \(1 + \cos\theta = 1 - \cos\theta\). Solve for \(\theta\) to find the limits of integration.
Recall that the area enclosed by a polar curve \(r(\theta)\) from \(\theta = a\) to \(\theta = b\) is given by the formula \(\frac{1}{2} \int_a^b r(\theta)^2 \, d\theta\).
Set up the integral for the area of the region inside \(r = 1 + \cos\theta\) but outside \(r = 1 - \cos\theta\). This will be the difference of the areas enclosed by the two curves between the intersection points: \(\frac{1}{2} \int_a^b \left[(1 + \cos\theta)^2 - (1 - \cos\theta)^2\right] d\theta\).
Evaluate the integral over the interval found in step 2 to find the area of the region between the two cardioids.

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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 angle (r, θ), which is essential for understanding curves like cardioids. Graphing these curves helps visualize the regions bounded by the given equations, making it easier to set up integrals for area calculation.
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Intro to Polar Coordinates

Area Between Curves in Polar Coordinates

The area between two polar curves is found by integrating the difference of their squared radii over the relevant angle interval. Specifically, the formula involves integrating ½(r_outer² - r_inner²) dθ, which accounts for the region inside one curve but outside the other.
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Intro to Polar Coordinates

Determining Limits of Integration

Identifying the correct angle bounds where the two cardioids intersect is crucial for setting up the integral. These limits ensure the integration covers only the region inside one cardioid and outside the other, avoiding overcounting or missing parts of the area.
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One-Sided Limits
Related Practice
Textbook Question

7–8. Parametric curves and tangent lines

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

x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6

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Textbook Question

10–12. Parametric curves

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

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

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Textbook Question

44–49. Areas of regions Find the area of the following regions.

The region inside the limaçon r=2+cosθ and outside the circle r=2

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Textbook Question

24–26. Sets in polar coordinates Sketch the following sets of points.


4 ≤ r² ≤ 9

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Textbook Question

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.


r = 3/(1 - 2 cos θ)

Textbook Question

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

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