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Do Over: Integration for Over Regions in the Plane

Introduction
Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons.
Why Double Integrals Again?
As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue.
The main cause of this issue was from the initial function that I chose. I had the function \(f(x,y)=2yx^2+9y^3\). Since this function included a squared and a cubed power the volume of the region came out to 4,811 cubed units. This is a HUGE number that had to be scaled down to even fit on the 3D printer. I have quite a few problems in front of me and I believe there is really only one solution.
The Solution
In my opinion, the only real way to resolve all of these issues and come up with a better integration for over regions in the plane model is to completely start from scratch. This was the best solution since the function I chose was not very friendly and caused a horrible 3D print. With the new function that I chose, I will be able to better represent a model for integration for over regions in the plane.
The Implementation
To begin, I need a completely new function that will not produce an exponentially large model. The function I chose that will hopefully fulfill this idea is \(f(x,y)=5+x^2-y^2\). In terms of the region for the rectangular to be enclosed by, I have chosen the region [0,2] x [0,2] with a step size of 0.5. This will produce roughly 36 rectangular prisms! Below is the approximation chart that contains the rectangular approximation method for right-hand endpoints.
Notice that there are 3 rectangular prisms with a negative approximation. These will not show up in our model. Here is the respective 3D model in openscad for my function.
As you can see at the front of the model, there are three prisms missing due to having a negative value! From the approximation, I approximated a value of 180 cubic units with an actual value of 125 cubic units. This was drastically less than my previous function. I believe that this example does a better job at visualizing integration for over regions in a plane. This model will print out far better than my previous model due to there being structural integrity. I also really like how the model has both increasing and decreasing values to show how interesting integration really is!

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