Section 8.2

Surface Area of Revolution

The Surface Area of a Solid of Revolution is essentially the "painted" area on the outside of the shapes we were finding the volume for earlier.

Think of it as taking the Arc Length formula and "spinning" it. As that tiny segment of the curve () rotates around an axis, it traces out a thin circular ribbon.

The total surface area is the sum of the areas of all these ribbons — just like wrapping a vase in tape!

The Surface Area Formula

How it Works: The Geometry

The area of one thin ribbon is its circumference multiplied by its length (the arc length):

The Radius (

)

The distance from the axis of rotation to the curve. This depends on which axis you rotate around.

The Arc Length (

)

The "walking distance" of that tiny segment:

Surface Ribbon

Unwrapping a surface band

Area = 2π * Radius * ArcLength

Like a label on a soup can

The Formulas

Depending on which axis you rotate around, the "radius" changes:

1. Rotation around the x-axis

The radius is the height of the function, .

2. Rotation around the y-axis

The radius is the horizontal distance from the axis, which is simply .

Note: Even when rotating around the -axis, you can still integrate with respect to as long as you use the correct radius ().

Step-by-Step Process

To solve a surface area problem, follow these steps:

Find the Derivative: Compute

Square and Add One: Set up

Identify the Radius:

Around -axis →
Around -axis →

Set up the Integral: Combine

, your radius, and your radical.

Evaluate: These integrals often require

-substitution.

Worked Example

Surface Area of a Sphere

Rotate (a semicircle of radius 2) about the -axis on .

Step 1: Find the Derivative

Step 2: Square and Add 1

Step 3: Take the Square Root

Step 4: Set up the Integral

Using the x-axis formula with :

Notice the beautiful cancellation!

Step 5: Evaluate

Answer:

This confirms the formula for a sphere with !

Level Up Examples

Example 2: Surface Area of a Cone

Rotate about the -axis on .

Step 1: Find the Derivative

Step 2: Square and Add 1

A constant — this makes integration easy!

Step 3: Set up and Integrate

Answer:

Example 3: Gabriel's Horn (A Paradox!)

Rotate about the -axis on .

Recall: The volume of Gabriel's Horn is finite (). What about the surface area?

Step 1: Set up the Integral

Step 2: Use Comparison Test

Since for all :

But we know diverges (it's a p-integral with )!

Answer: Surface Area is INFINITE!

This is the famous Painter's Paradox: Gabriel's Horn has finite volume but infinite surface area. You could fill it with paint, but you could never paint the outside!