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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 because some blocks would have negative volumes and some would have positive volumes. This caused some of the blocks to never touch because they only intersected at the corners, and all fell apart when I removed supports. It was as if, I printed floating blocks. This led me to change the domain, so that none of the individual prism volumes is below zero and all build up from a base and will have shared faces. I also think it would be beneficial to focus on a smaller portion of the function to provide a clear demonstration of the change in volume compared to my previous example that had a broad range and didn’t see much change.

The Improvement

To implement these improvements, I first adjusted the function to add and intercept, f(x) = cos(xy) +1. Then, I changed the domain to integrate on [0,3] x [0,3]. This allows for all the prisms to have a positive volume and build up from the base, so there will be no floating blocks. It also shows a single quadrant of the function to better show the volume change even though we know these volume changes occur similarly in all four quadrants. This resulted in a more accurate estimated volume compared to the actual volume, especially compared to my previous attempt. Below are my calculations to support these changes.

The estimated volume is 36.25 cubic units and represented in the table below. I used the top right corner of the prisms to calculate the individual prism volumes. 


The actual volume is 10.85 cubic unit which is solved by taking the double integral   


 






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