Algebra I / Elem. Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems
Algebra II / E&I Algebra
Exponents & Polynomials Intermediate Algebra starts here!

Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III
Calculus IV
15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis

Course: Calculus III
Topic: Vector-Valued Functions
Subtopic: Arc Length

Overview

As we study curves spun out in space by vector-valued functions, an obvious next step would be to find the length of that curve from one spot to another. In this lesson we find the arc length of a curve given by a vector-valued function or by a polar equation. The arc length of a curve can describe the distance an object travels on a trajectory in space over a specific time period. These concepts are natural extensions of arc length of 2D plane curves as studied in Calculus II.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 14.4.1 Find the arc length of a curve given by a vector-valued function
• 14.4.2 Write the arc length of a curve given by a vector-valued function as an integral and be able to evaluate that integral exactly or approximate it electronically
• 14.4.3 Given an object moving in a plane or in space with position vecr(t), find the distance it travels over a specific time interval
• 14.4.4 Find the arc length of a polar curve
• 14.4.5 Understand and be able to apply the arc length as a parameter theorem
• 14.4.6 Understand the term "parametrized by arc length" and be able to perform arc length parametrization
• 14.4.7 Recognize if a given vector-valued function has arc length as a parameter or not
• 14.4.8 Describe a curve using arc length as a parameter
• 14.4.9 Understand the physical, graphical, and algebraic connections of the statement "The rate of change of arc length with respect to time is the speed of the object moving on the curve" ((ds)/(dt) = ||vecv(t)||)

Terminology

Define: arc length, length of a curve, length of a polar curve, parameter, parametrized, parametrization, arc length parametrization

Formulae to have in your notes: arc length of a curve given by a vector-valued function; arc length of a polar curve; arc length as a function of parameter theorem

Supplemental Resources (optional)

Dale Hoffman's Contemporary Calculus III: Arc Length and Curvature of Space Curves (study "Arc Length" sections; save "Curvature" for a later lesson)

Paul's OL Notes - Calc III: Arc Length with Vector Functions

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vector Valued Functions". There are several related titles in the second half of that list.