We compute surface area of a frustrum then use the method of “Slice, Approximate, Integrate” to find areas of surface areas of revolution.

### The area of a frustum

In order to perform the approximation step, we first need to discuss the surface area
of a *frustrum*.

**frustum**of a cone is a section of a cone bounded by two planes, where both planes are perpendicular to the height of the cone.

To compute the area of a **surface of revolution**, we approximate that this area is
equal to the sum of areas of basic shapes that we can lay out flat. The argument for
this goes way back to the great physicist and mathematician, *Archimedes of
Alexandria*. To follow his argument, we have to begin by computing the area of a
‘lamp shade’ or *frustum*.

And of course, few things are more interesting than the area of a frustum:

- denote the number of trapezoids,
- denote the length of the top of each trapezoid,
- denote the height of each trapezoid,
- denote the length of the bottom of each trapezoid,

then from geometry, we have that each of the trapezoids, one of which is shown below:

- is the circumference of the top circle,
- is the slant height of the frustum as shown in the above figure, and
- is the circumference of the bottom circle,

then

and by way of limit laws we find Now, letting

- be the radius of the circle defining the top of the frustum,
- be the slant height of the frustum, and
- be the radius of the circle defining the base of the frustum,

we see that:

### The area of a surface of revolution

Let’s consider a function with a continuous derivative, and form a surface of revolution formed by this curve by rotating the portion of the curve from to about the -axis:

We can find a formula that gives the surface area of this surface of revolution using the procedure of “Slice, Approximate, Integrate”!

Step 1: Slice Since we have the curve to be revolved expressed as a function of , we choose to slice with respect to :

Step 2: Approximate We have seen how to find the surface area of a frustrum, so we should thus approximate each slice as a frustrum.

Thus the surface area, of this frustum is: Note that there is a value between and such that , so we write:

and can find the total approximate surface area by using frustra by adding together all of the surface areas:

Step 3: Integrate The formula above has good conceptual meaning, it does not readily pass to an integral quite yet! and have seen that we can express free in terms of either or , which allows us to express the infinitesimal by:

Note also that as the slice widths shrink, the value above approaches the distance that the corresponding slice is away from the axis of rotation.

To make sure that we emphasize this freedom in expressing as well as the inherent geometric results we used to build the surface area, we write:

where the radius is the distance from the axis of rotation to the slice and is the slant height of the slice.

To compute this surface area, we first choose to express either:

We then have to express the distance in terms of the variable of integration. This will always be a vertical or horizontal distance, which can be computed just as we have been doing in previous sections!

Using the remark, and letting and we can therefore write:

where the radius is the distance from the axis of rotation to the slice at .

An important concept to note is that the slice is located at a point on the curve. The choice of variable of integration may require that we express either or in terms of the other by using the equation that describes the curve. We will see this in the following examples.

We begin by considering looking at a picture,

As a brief aside, note that this slice gives rise to the following frustrum when revolved about the -axis:

Let’s first set up the integral with respect to . In order to do this, we choose:

Note that here is a vertical distance that must be expressed in terms of . Since the slice is located at a point on the curve :

Thus, .

Now, write with me: So

Note that here is a vertical distance that must be expressed in terms of . Since the slice is located at a point on the curve :

Thus, .

Now, since we have , write with me: So

This integral can be computed using the substitution . Working out the details (which you should do on your own), gives:

As our final example, we will compute the surface area of the sphere.

### Final thoughts

The key formulas in this section are:

We are free to choose the variable of integration here since we can express in terms of either or easily. Once this choice of variable has been determined, we need to express the radius of the infinitesimal frustrum and the limits for the integral in terms of the variable of integration.

This radius is the distance from the axis of rotation to the slice at , which is either a horizontal or vertical distance. We just need to make sure that we express it in terms of the variable of integration appropriately.

Many of the integrals that arise in the context of these problems can be difficult. Careful differentiation and algebra, as well as a good grasp of integration techniques can be vital when finding surface areas. As usual, this can be challenging and practice is the key here.

“Math is not a spectator sport. It’s not a body of knowledge. It’s not symbols on a page. It’s something you play with, something you do” - Keith Devlin