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...

Stereographic Projections : Stars

Stereographic Projections : Stars Previously, we have been discussing surfaces that are relatively easy to think about, and now we are going to consider a way to view a relatively simple surface, namely a sphere. We will be doing this via stereographic projections. But what is a stereographic projection? If you hear the word projection and immediately think of a (cartographic) map you are on the right track—as some maps are, in fact, stereographic projections. More formally, a stereographic projection is a mapping that projects a sphere onto a plane; so, by projecting a point(s) \(P\) on the surface of the sphere from the sphere's north pole \(N\) to point \(P'\) on a plane tangent to the sphere's south pole \(S\) as shown in the image below— note \(O\) denotes the origin. This allows use to view what is technically a three-dimensional image in two dimensions; which is very useful in cartography, geology, etc. However, there are ce...

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...

MA 391 Composition and Communication

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