I was doodling the other day when a thought occurred to me: “what is the area of a polynomial?” In particular, consider the following two graphs:
In the first graph, the blue line is straight. Algebraically, this is a first degree polynomial.
In the second graph, the blue line is curved. Algebraically, this is a parabola; a second degree polynomial.
The area of the first shape is common knowledge:
The area of the second shape can be calucated using calculus and is:
What if we considered a more complicated polynomial? A third degree polynomial looks like this:
The new curve bends even more and takes up more area. Using calculus, we find that the area is
Below are the graphs of 10th, 20th, and 100th degree polynomials:
The larger and larger degree polynomials start to look like a square! This means, that we would expect the area to get closer and closer to 1. This is exactly what I found. The area of a polynomial with degree 100 is 0.99. In general, the area of a polynomial with degree n is:
Taking the limit, we find that:
I found it very interesting that a large degree polynomial becomes so similar to a square. Click the link to view an animation displaying this fact https://www.desmos.com/calculator/afd21wuv0a