The Approximation of a Solid of Revolution

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is there a way that it can be visualized or approximated in such a way that makes it easier to understand than a verbal explanation?

Approximation Using Washers

The best way to construct a visual of how this solid looks is to approximate using a shape that has a volume that can be calculated fairly easily. The goal is to create disks with holes in the center to either wrap around or fit just inside the bounds defined by the function. These disks are known as washers. These washers have three relevant properties: height, inner radius, and outer radius. The inner and outer radii are defined by the function itself, regardless of the amount of partitions made, and will vary from washer to washer. Each radii will be defined by the respective function at the y-value of a partition. All of the washers will share the same height, and their height will be determined by the number of partitions that will be made for the function. For this example, there will be 10 partitions. This process is analogous to Riemann sums for finding area under a curve. In that case, as the rectangles become thinner and thinner, the more accurate the estimation is. In this case, the thinner the washer is the more accurate the 3-D model approximation will be.

For this example, the area that will be considered is the region bounded by the two functions on the range [0,1] along the y-axis. The easiest place to start is constructing the first washer. There will be 10 washers, meaning that each one will have a height .1. Now comes the process of finding the inner and outer radii. In this specific case, defines the size of the inner radius. A reliable visual way to determine this is to look at both functions and consider their valur at the point of interest, in this case that point is when y =.1. At this point, it can be seen on the graph that when y =.1, g(x) is closer to the y-axis. This indicates that the distance from that point to the y-axis would be hollow for the solid of revolution. The opposite applies for the other function, which is farther away from the y-axis.This indicates that the region between f(x) and g(x) is the region that is used to create the solid. This method can work even if the y-axis is not the axis of rotation; judge the distance from the relevant axis of rotation for each function. It is important to note that the value at f(x) determines the radius of the washer. Now that this information has been established, creating the washers is just a matter of plugging in the relevant values for each function.

Creating the Washers

For this example, the first washer's radii will the values of the function at y = .1. At this point the inner radius of the first washer is defined by the inverse of each function. The inverse of the functions are: \[ g^{-1}(y) = 4*ln(y+1)\] and \[ f^{-1}(y) = \frac{4ln(y+1)}{ln2} \] This will make finding the radii much easier as the desired y value can be plugged in for each function. For example, the height of the first washer has already been defined as 1. The inner radius is found by plugging in the value of y =.1 into the inverse of g(x): \[ g^{-1}(.1) = 4*ln(.1+1) = .38124\] Similarly, use the inverse of f(x) to find the outer radius: \[ f^{-1}(.1) = \frac{4ln(.1+1)}{ln2} = .55001\] So all of the properties needed have been defined and the first washer is done! At this point, The radii that need to be calculated can simply be plugged into each function. The following picture is a top view of the model approximation:

The highlighted washer is the tenth and final washer of the model.

Using the Model to Approximate Volume

So we have a model, now what? If the volume of each of these 10 washers is added up, the volume of the solid of revolution can estimated. This process is simple, but tedious, and will yield an estimation of the actual volume of the solid. The volume of a cylinder is: \[ V = \pi r^2h \] For each washer, the volume is calculated by plugging in the outer radius and subtracting by the volume of the inside cylinder. This method of approximation yields a volume of 11.5732 cubic inches. This model, when scaled 1:1 of inches to function value, will have a height of 1, and the largest cylinder will have a diamter of 8 inches. In order for this model to have a volume of around 1.5 cubic inches, the model would have to be scaled down by a factor of about 10.

Finding the Actual Volume of the Solid

Now that we have an approximation, a great question to ask is how good is it? To answer this question, integration must be used. Recall the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] The region is bounded above by g(x) and below by f(x) and the line y = 1. There is an argument to be made on whether to integrate the function with respect to x or y. I will carry out the integration with respect to x. To determine the bounds of integration for each function, the point at which f(x) = 1 and g(x) = 1 need to be found. The inverses derived earlier can be used to make this process quite easy. The bounds for the integral of f(x) are 0 and 4. The domain for g(x) is [0,2.7726]. Using the formula for integration using the washer method, the volume of the solid of revolution is defined by: \[ V = \int_{0}^{2.7726}\pi (e^{.25x} - 1)^2 \,dx - \int_{0}^{4}\pi (2^{.25x} -1)^2 \,dx = 15.34805 \] Most certainly not the prettiest integral, but the volume has been found.

Assessing the actual volume to the volume obtained from the approximation, there was an error of about 24.59%. The approximation accomplished its job decently and the image generated by the approximation provides a great deal of insight to the world of approximations of solids via washers.