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


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.


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


Terms you should be able to 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.