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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 12.3.65
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

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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.
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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.
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Arc 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.
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Intro to Polar Coordinates
Related Practice
Textbook Question

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

Textbook Question

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

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Textbook Question

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

Textbook Question

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 3 sin θ and r = 3 cos θ

Textbook Question

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.


Spiral of Archimedes: r = aθ

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Textbook Question

Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment. 

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