Textbook Question
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(1, 2π/3)
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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 12.3.84
84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is
L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.
Verified step by step guidance1
Recall the formula for the arc length of a curve given in Cartesian coordinates: if a curve is parameterized by \( x(t) \) and \( y(t) \) for \( t \) in \( [a, b] \), then the arc length \( L \) is \( L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \).
Express the polar curve \( r = f(\theta) \) in Cartesian coordinates using the relationships \( x = r \cos \theta = f(\theta) \cos \theta \) and \( y = r \sin \theta = f(\theta) \sin \theta \).
Compute the derivatives \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \) using the product rule: \( \frac{dx}{d\theta} = f'(\theta) \cos \theta - f(\theta) \sin \theta \) and \( \frac{dy}{d\theta} = f'(\theta) \sin \theta + f(\theta) \cos \theta \).
Substitute these derivatives into the arc length formula: \( L = \int_\alpha^\beta \sqrt{\left(f'(\theta) \cos \theta - f(\theta) \sin \theta\right)^2 + \left(f'(\theta) \sin \theta + f(\theta) \cos \theta\right)^2} \, d\theta \).
Simplify the expression inside the square root by expanding and combining like terms, using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), to show that it reduces to \( \sqrt{f'(\theta)^2 + f(\theta)^2} \), thus proving the formula \( L = \int_\alpha^\beta \sqrt{f(\theta)^2 + f'(\theta)^2} \, d\theta \).
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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 in the plane using a radius and an angle (r, θ). A polar curve is defined by a function r = f(θ), describing how the radius changes with the angle. Understanding this system is essential to relate the curve's shape to its parametric form.
Recommended video:
05:32
Intro to Polar Coordinates
Arc Length Formula for Parametric Curves
The arc length of a curve defined parametrically by x(t) and y(t) from t = a to b is given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula generalizes the distance traveled along a curve and is the foundation for deriving arc length in polar form.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation and Chain Rule in Polar Coordinates
To find the arc length of a polar curve, one must differentiate r = f(θ) and convert the curve into parametric form x = r cos θ, y = r sin θ. Applying the chain rule to these expressions allows computation of dx/dθ and dy/dθ, which are used in the arc length integral.
Recommended video:
05:32Intro to Polar Coordinates
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 = √t + 4, y = 3√t; 0 ≤ t ≤ 16
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
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.
10x² - 7y² = 140
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Textbook Question
53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.
The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.
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