Most of the projects I have posted on here have been rather abstract. I want to connect what I do with reality more often (although math is the very essence of the universe).
In mathematics, we often “cheat” by going as close to infinity as possible and pulling out at the last second. This can be seen in integrals, derivatives, and Taylor series. All of these make use of infinitely small or infinitely large values. However, most functions cannot be integrated by hand and most Taylor series cannot be calculated to an infinite degree. As a result, we often rely on computers to calculate these values with brute force, that is, calculating the Riemann sum in the case of the definite integral or finding the Taylor series to a certain degree.
“Real” Riemann Sums
A while back, I made a post (link) regarding Riemann sums. By using Mathematica, I was able to instantly generate hundreds of rectangles using left, right, and midpoint rule. Instead of relying on unrealistically large numbers of rectangles for my Riemann sum, I decided to go with 29 slices on a relatively large interval. I used the function shown below on the interval [1,6.25].
Instead of doing a 2-dimensional Riemann sum, I did a volume integral so a 3-dimensional model could be physically constructed. The Riemann sum was composed of squares perpendicular to the x-axis. I planned to cut out squares from foam board and glue them together to make a model of the function. The thickness of the foam board was 3/16″, and 1 unit corresponded to 1 inch.
All of the calculations were done in Google Sheets (link to the original document).
As you can see, the final Riemann sum, shown as the cumulative volume, was an overestimate, and the error was 3.6406. This number is surprisingly accurate of 27 slices on a large interval. The accuracy of the function can really be seen when the squares (which literally took hours to cut out) are glued together. The width of each slice is represented as f(x) on the table.
Here are all of the slices laid out.
Gluing in process. I really should have used wet glue, but everything worked out.
…And finally, here are two views of the final product! Even with extreme variation in the beginning of the interval, the model turned out fairly accurate. It is now easy to imagine how more complex models can be constructed with high accuracy in more professional settings, such as when a satellite is constructed.
**NOTE: All of the equations were done in a interpreter, because I cannot directly put nice-looking equations in WordPress.