Textbook Question
What is the polar equation of the horizontal line y = 5?
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.3.33
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the curve r = √(cos θ)
Verified step by step guidance1
First, understand the curve given in polar coordinates: \(r = \sqrt{\cos \theta}\). Since \(r\) must be real, identify the domain of \(\theta\) where \(\cos \theta \geq 0\). This will help determine the bounds for integration.
Sketch the curve by plotting points for values of \(\theta\) within the domain found. Note that \(r\) is non-negative and depends on \(\cos \theta\), which is positive in the intervals \([-\frac{\pi}{2}, \frac{\pi}{2}]\) and \([\frac{3\pi}{2}, \frac{5\pi}{2}]\), but since \(r\) is defined as \(\sqrt{\cos \theta}\), focus on the principal interval where \(r\) is real and positive.
Recall the formula for the area enclosed by a polar curve between angles \(\alpha\) and \(\beta\) is \(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). Substitute \(r^2 = \cos \theta\) into this formula.
Set up the integral for the area as \(A = \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos \theta \, d\theta\), using the interval where \(r\) is real and positive.
Evaluate the integral by integrating \(\cos \theta\) over the interval, then multiply by \(\frac{1}{2}\). This will give the area of the region inside the curve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Curves
Polar coordinates represent points using a radius and an angle (r, θ). Curves defined in polar form, like r = √(cos θ), describe shapes based on these parameters. Understanding how to interpret and plot such curves is essential for visualizing the region whose area is to be found.
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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 the integral (1/2) ∫[a to b] (r(θ))^2 dθ. This formula accounts for the sector-like slices of the region, making it crucial for finding areas bounded by polar curves.
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05:32Intro to Polar Coordinates
Determining the Limits of Integration
To compute the area, one must identify the correct interval [a, b] for θ where the curve is defined and the region is enclosed. This often involves analyzing where r(θ) is real and non-negative, and where the curve completes a full loop or returns to the starting point.
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05:50One-Sided Limits
Related Practice
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 = r − 1, y = r³; −4 ≤ r ≤ 4
Textbook Question
Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)
Textbook Question
Given three polar coordinate representations for the origin.
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Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
x² = 12y
Textbook Question
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete cardioid r = 4 + 4 sin θ