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

We have already seen how a curve described by $y=f(x)$ on $[a,b]$ can be revolved around an axis to form a solid. Another geometric quantity of interest is the surface area of this solid. We will be able to apply the procedure of “Slice, Approximate, Integrate” to find the surface area.

The area of a frustum

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

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

Now we are ready to compute the area of a surface of revolution.

The area of a surface of revolution

Let’s consider a function $f$ with a continuous derivative, and form a surface of revolution formed by this curve by rotating the portion of the curve from $x=a$ to $x=b$ about the $x$-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 $x$, we choose to slice with respect to $x$:

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, $\Delta SA$ of this frustum is: Note that there is a value $r$ between $r_1$ and $r_2$ such that $\frac {r_1+r_2}{2} = \frac {2r}{2} = r$, so we write:

and can find the total approximate surface area by using $n$ 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 $\Delta s$ free in terms of either $\Delta x$ or $\Delta y$, which allows us to express the infinitesimal $\d s$ by:

Note also that as the slice widths shrink, the value $r$ 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 $\d s$ as well as the inherent geometric results we used to build the surface area, we write:

Using the remark, and letting $A=(a,c)$ and $B=(b,d)$ we can therefore write:

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

The final answers in the previous two examples are:
equal not equal

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 $\d s$ in terms of either $\d x$ or $\d y$ easily. Once this choice of variable has been determined, we need to express the radius $r$ of the infinitesimal frustrum and the limits for the integral in terms of the variable of integration.

This radius $r$ is the distance from the axis of rotation to the slice at $(x,y)$, 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