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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh.12 - Parametric and Polar Curves문제 12.3.53
12장, 문제 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

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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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
05:32
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.
추천 영상:
05:32
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.
추천 영상:
가이드 코스
09:04
Slope of Polar Curves
관련 실천
교과서 질문

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

74
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교과서 질문

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


(1, √3)

47
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교과서 질문

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

46
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교과서 질문

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?

45
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교과서 질문

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

52
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교과서 질문

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

r= √(cos3t) and r= √(sin3t)

68
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