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Welcome to MA 391: Composition and Communication at the University of Kentucky!

This semester we are going to be practicing the (sometimes frustrating and always ) important skill of communicating mathematics. At the start of the semester we are going to revisit some topics from calculus and for each of those topics you will pick an example you think really helps explain what is going on. You will design and print a 3d model of that example. Then you will write a blog post (here!) and a description card for your model and participate in the class show and tell.

After spending some time in calculus we will switch over to topics motivated by geometry and topology. You don't need to know anything about these area yet. We're going to look at them since they will be new to many of you, they lend themselves well to visualization projects and my research is in topology!

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Finding an object's center of mass using integration

Thinking about center of mass As a kid, I would sit in class and balance my pencil on my finger. What I didn't know then was that I was finding the pencil's center of mass. An object's center of mass is a point where the average weight of the object lies. In other words, the weighted position vectors of the object at this point sum to zero (source: https://www.dictionary.com/browse/center-of-mass ). The exact location of this point can be found using calculus. First, it is important to understand the moment . The moment for a two-dimensional object is given with respect to both the x- and y-axes. It is a physical characteristic of the object a certain distance from these axes. It's easiest to think about the axes as reference points for where the characteristic acts; in this case, where the mass is distributed. When computing these moments, the hardest concept to grasp is that the moment with respect to y, for example, is given by the x-coordinat...
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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...
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Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
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