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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 12.3.53
Chapter 12, Problem 12.3.53

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


The region common to the circles r = 2 sin θ and r = 1

Verified step by step guidance
1
First, understand the problem: we need to find the area of the region common to the two circles given in polar coordinates: \(r = 2 \sin \theta\) and \(r = 1\).
Step 1: Sketch or visualize the two circles to understand their intersection points and the region they enclose. The circle \(r = 2 \sin \theta\) is a circle centered at \((0,1)\) with radius 1, and \(r = 1\) is a circle centered at the origin with radius 1.
Step 2: Find the points of intersection by setting the two equations equal: \(2 \sin \theta = 1\). Solve for \(\theta\) to find the limits of integration for the common region.
Step 3: Set up the integral for the area of the common region. The area enclosed by a polar curve \(r(\theta)\) from \(\alpha\) to \(\beta\) is given by \(\frac{1}{2} \int_{\alpha}^{\beta} r(\theta)^2 \, d\theta\). For the common region, the area will be the sum of areas bounded by each curve between the intersection angles, taking care to integrate the smaller radius where appropriate.
Step 4: Write the integral expressions for the area of the common region by splitting the integral at the intersection points and using the appropriate \(r(\theta)\) for each part. Then, prepare to evaluate these integrals to find the total area.

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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 an angle (r, θ). Understanding how to graph polar equations like r = 2 sin θ and r = 1 helps visualize the regions defined by these curves, which is essential for setting up the area calculation.
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Intro to Polar Coordinates

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) between angles α and β is given by (1/2) ∫ from α to β of [r(θ)]² dθ. This formula is crucial for finding the area of regions bounded by one or more polar curves.
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Intro to Polar Coordinates

Finding Intersection Points of Polar Curves

To find the common region between two polar curves, you must determine their points of intersection by solving r₁(θ) = r₂(θ). These intersection angles define the limits of integration for the area calculation.
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Slope of Polar Curves
Related Practice
Textbook Question

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(1, √3)

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Textbook Question

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

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Textbook Question

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

b. At what point does the tangency occur?

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b

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Textbook Question

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

r= √(cos3t) and r= √(sin3t)

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