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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 12.3.45
Chapter 12, Problem 12.3.45

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


The region outside the circle r = 1/2 and inside the circle r = cos θ

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First, understand the problem: we want to find the area of the region that lies outside the circle given by \(r = \frac{1}{2}\) and inside the circle given by \(r = \cos \theta\) in polar coordinates.
Next, find the points of intersection between the two circles by setting their equations equal: \(\frac{1}{2} = \cos \theta\). Solve this equation for \(\theta\) to determine the limits of integration.
Recall that the area enclosed by a polar curve \(r = f(\theta)\) between angles \(\alpha\) and \(\beta\) is given by the integral \(\frac{1}{2} \int_{\alpha}^{\beta} (f(\theta))^2 \, d\theta\). We will use this formula to find the areas inside each circle.
Set up the integral for the area inside \(r = \cos \theta\) but outside \(r = \frac{1}{2}\) by subtracting the area inside \(r = \frac{1}{2}\) from the area inside \(r = \cos \theta\) over the interval between the intersection points found in step 2.
Finally, evaluate the integral expressions (or leave them in integral form if not asked to compute) to express the area of the desired region.

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Key Concepts

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

Polar Coordinates and Graphs

Polar coordinates represent points using a radius and an angle (r, θ). Understanding how to graph and interpret curves like r = 1/2 and r = cos θ is essential to visualize the regions bounded by these curves.
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Intro to Polar Coordinates

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by (1/2) ∫[a to b] (r(θ))^2 dθ. This formula is fundamental for finding areas of regions defined by polar curves.
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Intro to Polar Coordinates

Determining Intersection Points and Limits of Integration

To find the area between two polar curves, it is crucial to identify their points of intersection by solving r1 = r2. These intersection angles set the limits for integration when calculating the enclosed area.
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Determining Different Coordinates for the Same Point
Related Practice
Textbook Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = sin θ sec² θ

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Textbook Question

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 1 and r = 2 sin 2θ

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Textbook Question

Explain why the slope of the line θ=π/2 is undefined.

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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 = √t + 4, y = 3√t; 0 ≤ t ≤ 16

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Textbook Question

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


The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant

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Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2

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