Textbook Question
Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)
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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 12.3.65
63–74. Arc length of polar curves Find the length of the following polar curves.
The spiral r = θ², for 0 ≤ θ ≤ 2π
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a curve given in polar coordinates \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \):
\[
L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d}{d\theta}r(\theta)\right)^2} \, d\theta
\]
Identify the given function and interval: here, \( r(\theta) = \theta^2 \) and \( \theta \) ranges from 0 to \( 2\pi \).
Compute the derivative of \( r(\theta) \) with respect to \( \theta \):
\[
\frac{d}{d\theta}r(\theta) = \frac{d}{d\theta}(\theta^2) = 2\theta
\]
Substitute \( r(\theta) \) and its derivative into the arc length formula to get the integrand:
\[
\sqrt{(\theta^2)^2 + (2\theta)^2} = \sqrt{\theta^4 + 4\theta^2}
\]
Set up the integral for the arc length:
\[
L = \int_0^{2\pi} \sqrt{\theta^4 + 4\theta^2} \, d\theta
\]
This integral can then be evaluated (using appropriate methods) to find the length of the spiral.
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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). Polar curves are defined by equations relating r and θ, such as r = θ². Understanding how to interpret and plot these curves is essential for analyzing their properties, including length.
Recommended video:
05:32
Intro to Polar Coordinates
Arc Length Formula for Polar Curves
The arc length of a polar curve r(θ) from θ = a to θ = b is given by the integral ∫ₐᵇ √[r(θ)² + (dr/dθ)²] dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length by integrating over the specified interval.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, you must compute the derivative dr/dθ of the polar function r(θ). This involves differentiating r = θ² with respect to θ, which yields dr/dθ = 2θ. Accurate differentiation is crucial for correctly evaluating the integral for arc length.
Recommended video:
05:32Intro to Polar Coordinates
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
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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Textbook Question
23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.
A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.
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Textbook Question
9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.
(-1, -π/3)
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