A Peculiar Property of Definite Integrals // The King Rule (The Art of Integration)

In this video we introduce a peculiar property of definite integrals that's known in some places as the King Rule. On our journey to evaluate difficult integrals we've needed to expand our mathematical toolbox to be able to increase our creative problem solving skills. We've used algebraic tricks, creative substitutions, and advanced results like Euler's formula and the gamma function. We're now ready to introduce a simple property of definite integrals that's incredibly useful for a large class of integrals. At first glance this property doesn't seem to be of much use. However, we'll see just how useful it is after going through two good introductory problems. Timestamps for the different parts are below: 0:00 Introduction 1:49 Geometric Meaning 5:51 Proof 10:43 Alternate Version 13:19 Problem 1 16:59 Problem 2 The Art of Integration is an ongoing series where we evaluate integrals with techniques that are not typically taught in the calculus sequence. This is a great way for students in science, engineering, and mathematics to strengthen their integration skills and creativity in solving problems. Most of the problems should be accessible to students that have covered the integration methods from calculus 2. Looking for a specific problem or topic? Try checking my website: https://www.blacktshirtmathprofessor.... ► Artist Attribution Music By: "After The Fall" Track Name: "Pieces" Music Published by: Chill Out Records LLC License: Creative Commons Attribution 4.0 International (CC BY - 4.0) Full license here: https://creativecommons.org/licenses​