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:

**Terminology**

Define: solid of revolution, surface, surface area

**Supplemental Resources (optional)**

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

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