Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
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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: Area of Surface of Revolution

Overview

In this lesson we will use definite integrals to find the area of the surface of a solid of revolution. If the solid is a regular shape, such as a cylinder, then we have a formula to find its surface area. However, if the curve is spun to generate the solid is irregular, then calculus is needed to find its surface area. The plan will be, as usual, to find the area of a tiny piece of the surface then add the little areas up to get the surface area.

Objectives

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

• 6.6.1 Understand how to find the radius r(x) or r(y) and when to use x vs. y
• 6.6.2 Set-up and evaluate the integral of 2πrds (in terms of x or in terms of y) that gives the surface area of a solid of revolution
• 6.6.3 Know the two surface area formulae and when to use which one:
SA=2 pi int_a^b r(x) sqrt(1+(f'(x))^2) dx
SA=2 pi int_c^d r(y) sqrt((f'(y))^2+1) dy

Terminology

Terms you should be able to define: solid of revolution, surface, surface area

Mini-Lectures and Examples

Supplemental Resources (optional)

Video: Arc Length and Surface Area, Selwyn Hollis's Video Calculus (watch 2nd half)

Lesson: Area of a Surface of Revolution, Math24's Calculus has good sketches related to SA formulae, but ignore the #3 and #4 examples in each section (we will study parametric and polar surfaces later).

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

rev. 2021-01-16