In this lesson, you will learn about arc length for y = f(x), arc length for vector function, arc length as a function of a parameter, and its applications.
View all of the instructional videos, read the provided readings, and complete practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided.
Upon completion of lesson 3.2, you will be able to:
Determine arc lengths for functions of single variable
Construct arc length parameterization
Model and solve applications
View all of the following instructional videos. These will help you master the objectives for this lesson.
Arc Length – Part 1 of 2 [7:08]
Source: bullcleo1 from YouTube
Arc Length – Part 2 of 2 [4:52]
Source: bullcleo1 from YouTube
Arc Length – Find the arc length of a function [10:00]
Source: patrickJMT from YouTube
Arc Length Using Parametric Curves - Clip 1 [8:17]
Source: patrickJMT from YouTube
Arc Length Using Parametric Curves - Clip 2 [7:27]
Source: patrickJMT from YouTube
Arc Length of a Parametric Curve, Another Example #1 [1:50]
Source: patrickJMT from YouTube
Arc Length of a Parametric Curve, Another Example #2 [5:23]
Source: patrickJMT from YouTube
Determining Arc Length of a Curve Defined by a Vector Valued Function [7:42]
Source: bullcleo1 from YouTube
Note: If the curve is not a smooth curve, as long as it is a piece-wise smooth curve, you can apply the arc length formula to find the length of each smooth piece, then add all the lengths together. In Example 2, instead of integrating on the interval [0, π], the presenter breaks the interval into two parts, [0, π/2] and [π/2,π], because the curve on the interval [0,π] is not smooth, however, it consists of two pieces of smooth curves, the curves traced out while the parameter varies on the interval [0, π/2] and on the interval [π/2,π].
Arc Length of a Vector Function [7:42]
Source: patrickJMT from YouTube
Note: I only found these two videos that cover the topic, arc length parameterization. They are not perfect, but I hope they are helpful.
Arc Length Parameterization [7:07]
Source: theschoolfchuck from YouTube
Note: The presenter made a simple math error at the end: t should be equal to s/5, not 5/s.
-7-motion-04-Arc Length Parameter [4:27]
Source: Ben Woodruff from YouTube
Note: Resolution is poor, but watchable.
The following required readings cover the content for this lesson. As you go through each reading, pay close attention to the content that will help you learn the objectives for this lesson.
HMC Mathematics Online Tutorial
Paul’s Online Math Notes
This document covers materials from lessons in both Module 2 and Module
Read slides 3-14 for this lesson.
Read slides 15-42 for Lesson 2.3.
Read slides 43- 66 for Lesson 3.1.
Read slides 90-102 for Lesson 3.2.
Read slides 67-89, 103-127 for Lesson 3.3.
Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.
Go to the Homework Hints, click on the Vector Functions, and choose Arc Length and Curvature.
Below are additional resources that help reinforce the content for this lesson.
Second half of this article highlights what you need to learn for the next lesson.
Deriving the formula for the length of an arc (video) [9:28]
Source: anortonMovies
Do Homework 3.2 on MyMathLab
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