Skip to main content

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

Comments

Post a Comment

Popular posts from this blog

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
8 comments
Read more

Finding the Center of Mass of a Toy Boat

Consider two people who visit the gym a substantial amount. One is a girl who loves to lift weights and bench press as much as she possibly can. The other is a guy who focuses much more on his legs, trying to break the world record for squat weight. It just so happens that these two are the same height and have the exact same weight, but the center of their weight is not in the same part of their body. This is because the girl has much more weight in the top half of her body and the boy has more weight in the bottom half. This difference in center of mass is a direct result of the different distributions of mass throughout both of their bodies. Moments and Mass There are two main components to finding the center of mass of an object. The first, unsurprisingly, is the mass of the whole object. In this case of the boat example, the mass will be uniform throughout the entire object. This is ideal a majority of the time as it drastically reduces the difficulty
Post a Comment
Read more