Ambiguous objects are in the display case!

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

Come visit the quadric surfaces!

They are in the display case. The ruled surfaces will be joining them soon!

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 o

MA 391 Composition and Communication

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