4.9 Arc Length

4.9 Arc Length

Summary

Another illustration of the Riemann Sum modeling method is given, this time to compute the length of a curve in the plane. An integral formula is developed to compute the arc length. It is pointed out that the formula often leads to integrals that must be approximated by numerical methods.

By the end of your studying, you should know:

The arc length formula.
How to use the arc length formula.
How to evaluate the arc length integral using the Fundamental Theorem if possible, and if not, using a numerical method (either by hand or with an applet).

On-screen applet instructions: Use the pull-down menu to change the value of n, the number of line segments.

Examples

Find the length of the parabola y = x² from x = 0 to x = 1.

Find the length of the curve y = sin(x) from x = 0 to x = π.

Find the circumference of the hypocycloid

Applets

Numerical Integration

Videos

See short videos of worked problems for this section.

Quiz

Exercises

See Exercises for 4.9 Arc Length (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Information on Leibniz
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Arc Length
Visual Calculus

Interesting Application

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4.8 Volumes of Solids of Revolution

Table of Contents

4.10 Inverse Trigonometric Functions

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel