For my do-over blog, I thought about the prints that I have done this semester. Of all my prints, the one I liked the least was the first one. Due to my lack of experience with the software and 3D printing process, the resulting product was lackluster in my eyes. With this iteration, I wanted to take a look at the same functions I chose the first time around, and see if choosing a different domain would create an object that was more like the shape I desired. The goal was and is to create a bowl shaped object that looks good on my desk.
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...
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