# 5.4: Area of a Surface of Revolution

## Learning Objectives

A student will be able to:

- Learn how to find the area of a surface that is generated by revolving a curve about an axis or a line.

In this section we will deal with the problem of finding the area of a surface that is generated by revolving a curve about an axis or a line. For example, the surface of a sphere can be generated by revolving a semicircle about its diameter (Figure 19) and the circular cylinder can be generated by revolving a line segment about any axis that is parallel to it (Figure 20).

**Figure 19**

**Figure 20**

**Area of a Surface of Revolution**

If is a smooth and non-negative function in the interval then the surface area generated by revolving the curve between and about the axis is defined by

Equivalently, if the surface is generated by revolving the curve about the axis between and then

**Example 1:**

Find the surface area that is generated by revolving on about the axis (Figure 21).

*Solution:*

**Figure 21**

The surface area is

Using substitution by letting

**Example 2:**

Find the area of the surface generated by revolving the graph of on the interval about the axis (Figure 22).

*Solution:*

**Figure 22**

Since the curve is revolved about the axis, we apply

So we write as . In addition, the interval on the axis becomes Thus

Simplifying,

With the aid of substitution, let

## Multimedia Links

For video presentations of finding the surface area of revolution **(16.0)**, see Math Video Tutorials by James Sousa, Surface Area of Revolution, Part 1 (9:47)

and Math Video Tutorials by James Sousa, Surface Area of Revolution, Part 2 (5:43).

## Review Questions

In problems #1 - 3 find the area of the surface generated by revolving the curve about the axis.

In problems #4–6 find the area of the surface generated by revolving the curve about the axis.

- Show that the surface area of a sphere of radius is .
- Show that the lateral area of a right circular cone of height and base radius is

## Review Answers

- We can create a sphere of radius by rotating a semicircle of radius around the axis. The formula for a circle of radius is , and we can solve for to get the equation for the semicircle: . Then we can also solve for: and so . Now we can plug all this into the integral that gives us the surface area, integrating from :

- We can create this right circular cone by rotating a line around the axis over the interval . Two points we know that have to be on the line are and , which then gives us an equation for the line of . Then we can determine the derivative of with respect to , and so . And now we are ready to calculate the integral that gives the lateral surface area: