Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
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Calculus I
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Calculus II
Transcendental Functions
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Calculus III

Course: Calculus II
Topic: Geometry
Subtopic: Arc Length

Overview

In this lesson we will use definite integrals to find the length of an arc, that is, the length of a portion of a curve. If the curve is a regular shape, such as an ellipse, then we have a formula to find its perimeter. However, if the curve is irregular then calculus is needed to find its length. The plan will be, as usual, to find the length of a tiny piece of the curve then add the little lengths up to get the arc length.

Objectives

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

• 6.5.1 Understand what "ds" is (a little piece of arc length) and how it relates to dx and dy via the Pythagorean theorem
• 6.5.2 Set-up and evaluate the integral of ds (in terms of x or in terms of y) that gives the arc length of a curve
• 6.5.3 Know the two arc length formulae and when to use which one:
AL = int_a^b sqrt(1+(f'(x))^2) dx
AL = int_c^d sqrt((f'(y))^2+1) dy

Terminology

Terms you should be able to define: arc of a curve, central angle, subtends (as in "the angle subtends the arc")

Mini-Lectures and Examples

Supplemental Resources (optional)

Video: Arc Length and Surface Area, Selwyn Hollis's Video Calculus (watch 1st half)

PDF: Lengths of Curves and Areas of Surfaces of Revolution, Dale Hoffman's Contemporary Calculus (study arc length sections)

Desmos Exploration: Arc Length

rev. 2021-01-09