A Calculus Proof That Traps Euler’s Number (e)

追加: 2026-05-16

[00:00:00]Want to perfectly trap Euler's number between two rational fractions? Here's a beautiful calculus trick using a single elegant integral from zero to one.

[00:00:08]First, expand the polynomial and apply integration by parts. When we evaluate the bounds, the entire integral simplifies cleanly to 14e - 38. Now, look at the graph. The integrand stays strictly positive between zero and one.

[00:00:22]So, the total area under the curve must be greater than zero. With a little algebra, we get our first result. E is strictly greater than 19/7.

[00:00:31]Next, we find an upper bound by overestimating the area. Since e to the x is always less than e on this interval, we can pull an e outside the integral. The remaining polynomial integrates to 1/30, giving an upper estimate of e over 30. That means the exact area, 14e - 38, must be less than e over 30.

[00:00:53]Solving this inequality shows that e is perfectly trapped between 19/7 and 1140/419.