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.