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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.3.45
Capitolo 12, Problema 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

Guida verificata passo dopo passo
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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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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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05:32
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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6:02
Determining Different Coordinates for the Same Point
Pratica correlata
Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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

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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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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