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 :

### 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