I “Solved” This Impossible Integral! 🧠
The integral of e^x^2 has no known elementary function solution, but in this calculus II video, we're going to challenge what a "solution" really means. While you can't use integration by parts or u substitution on this integral, we're going to get creative.
On your calculus BC exam and especially in the real world, sometimes you may have to approximate solutions when no "real" solution exists, and we'll do that here with a Taylor series centered at a=0. Theoretically, any function can be represented by this infinite series and for real world application, that may not be fully applicable at infinity. However, we'll demonstrate how even taking just a few of the series terms can give a real good approximation for the area under the curve! Hope you enjoy!
0:00 - Introduction
0:17 - No "Real" Solution
1:42 - Taylor Series Approximation
3:10 - Finding Coefficients (e^y)
4:29 - Finding Coefficients (e^x^2)
5:05 - Real World Application
Пікірлер: 11
What method did you use to approximate the integral?
No worries just define the error function as the solution to this integral 😂
@NumberNinjaDave
16 күн бұрын
Hahahahaha
@numbers93
15 күн бұрын
NGL I thought this was gonna be a 1 min prank vid and that he was gonna do exactly this 😄
@NumberNinjaDave
15 күн бұрын
@@numbers93 lol
I used this method specifically to prove the 68.2% 95.4% and 99.7% of the population under a gaussian within 1,2 and 3 standard deviations respectively, although a taylor expansion solution isn't as nice as an elementary function since it's hard to do things like solve for a value or find the inverse function for example
@NumberNinjaDave
16 күн бұрын
Oh that’s epic! I’ve never used this in a statistical application
@elibrahimi1169
16 күн бұрын
@@NumberNinjaDave well when i had to i had no choice but to use it, and there we go
Yeah but this is, just as you said, an approximation. Besides you can evaluate the integral in the video from 0 to 1 with the same method as the gaussian integral because both are converging to a finite value over the limits of integration. You can take I as the integral from -1 to 1 of e^x² and since e^x² is an even function, your target integral will be I/2. Then you can use any of the methods used to solve the gaussian just with the limits being -1 to 1 instead of -infinity to infinity
@NumberNinjaDave
17 күн бұрын
Very true, 🥷