Polynomial Area

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:

poly1  poly3

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:

\frac{1}{2} * base * height = \frac{1}{2} * 1 * 1 = \frac{1}{2}

The area of the second shape can be calucated using calculus and is:

\frac{2}{3} * base * height = \frac{2}{3} * 1 * 1 = \frac{2}{3}

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  \frac{3}{4}

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:

\lim_{x \to \infty} \frac{n}{n+1} = 1

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


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