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Do-Over: Integration Over Regions in the Plane

How I chose which project to repeat and what needs fixed

Over the course of the semester, I've learned a lot about not only explaining math but also the logistics of printing a helpful 3D model. In many of my calculus courses, professors have stood at the chalkboard and tried to draw a 3D object, only for the students to be more confused than before the visual aid was added. When I thought about which project to do again, I tried thinking of a project where my print was difficult to visualize and only made the topic more confusing. My integration over regions in the plane project came to mind.

Originally, I picked the function \(f(x, y) = \frac{1}{3}(x^2)(2y)\) on the plane bounded by \(y = e^{x-5}\) and \(y = ln(x)+5\) from [0, 7] x [0, 7]. This function grows very quickly, which was seen on my model because I picked such a large rectangular region. The print was very tall and super skinny. It was difficult to see the rectangles of the approximation because of this.
To fix this issue, I can do one of two things: 1) I can have the rectangular region enclosed by [0, 1] x [0, 1] so that the steps between x- and y-values are not so drastic, or 2) I can pick a new function. If I make the rectangular region smaller, you will no longer be able to see the shape of the domain, below.
Therefore, I am going to pick a function that does not grow as quickly over the same domain. This way, you'll be able to compare the original function to a better function and how in this new case you can see the details I originally intended for you to see much clearer. (I wanted the domain to have a leaf-like shape for fall, which I still want to celebrate Taylor Swift's new album.)

How I implemented the improvements

I started implementing these changes by first finding a slow-growing function of two variables that wasn't too basic. I ended up choosing \(f(x,y)=\frac{10}{2x+y}\).
As you can see, this function fixes my huge growth issue over the chosen region, and we can still see the shape of the domain. It may be on the less-interesting side of things, but any fancy functions with exponentials, etc. were not great for printing. Here is the model using the rectangular approximation method and right-hand endpoints:
As you can see, the steps are not as extreme, which was my goal. I think this model does a much better job at helping students visualize a rectangular approximation, given you can actually see the rectangles this time. My print is going to be blown up to have about a 3.5 x 3.5 inch base so that you can see everything as clearly as possible.

Word count: 468

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