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Solids of Revolution Revisited

Introduction

In my previous blog post on solids of revolution, we looked at the object formed by rotating the area between \(f(x) = -\frac{1}{9}x^2+\frac{3}{4} \) and \( g(x)=\frac{1}{2}-\frac{1}{2}e^{-x} \) around the \( x \) axis and bounded by \( x = 0 \) and \( x = 1.5 \). When this solid is approximated using 10 washers, the resulting object looks like this:

When I was looking back over the 3D prints I’d created for this course, I noticed that the print for this example was the least interesting of the bunch. Looking at the print now, I feel like the shape is rather uninteresting. The curve I chose has such a gradual slope that each of the washers are fairly similar in size and causes the overall shape to just look like a cylinder. Since calculating the changes in the radiuses of the washers is a big part of the washer method, I don’t think this slowly decreasing curve was the best choice to illustrate the concept. The reason I had done this originally was because I hadn’t realized you could add supports to the print, and so I wanted to avoid printing anything that would have an overhang. Now that I have a lot more experience with 3D printing, I am going to alter this example to fix these issues.

A New Example

To revise my example, I want to change the outside curve to something that changes directions, which will make the outside of the object much more exciting. I don’t want to use any trigonometric curves that would be difficult to integrate, so instead I’ve decided to use the 3rd degree polynomial, \( f(x)=x^3-4x^2+4x+1 \).

I considered whether or not to change the function for the inner radius as well, which was also a fairly unexciting curve. Ultimately, I decided to keep the same equation (although I did need to scale it fit my new boundaries) because together the resulting solid of revolution looked like a flower vase. Something I’d really liked about my original item was that it looked like a real world object (I compared it to a piece of candy), so I wanted to make sure that my new object still had some sort of application. I want it to look interesting while still having the appearance of an everyday object- in this case, a flower vase.

As a result, I ended up using the following curves on the interval \( 0\le x\le2.5 \) for my new example:

The resulting print is shown below and is much more eye-catching than my original. I think that this change made the object a much better representation of the washer method as well. The washer method can calculate the volume of so many different solids, so I think it is much more fitting for it to be demonstrated through a less boring example.

Author: Sarah Bombrys

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