The integral of x^n (a) | Famous Math Problems 10 | NJ Wildberger скачать в хорошем качестве

The integral of x^n (a) | Famous Math Problems 10 | NJ Wildberger

The integral of x^n is one of the two most important formulas in calculus. The cases n=0,1 and 2 go back to antiquity, and to Archimedes quadrature of a parabola. The general case was first enunciated by Cavalieri, who gave geometrical arguments for n=3,4,...,9. Subsequently Wallis, Fermat, Roberval and Torricelli gave arguments for the general case. However the problem is not at all completely solved. The issue is with the proper definition of the integral; here many problems of modern analysis emerge, having to do with the nature of the continuum (`real numbers'), infinite processes and limits, ambiguity about a prior theory of areas, and many sensitive technical details which are usually finessed in modern treatments. In this video and the next we outline a new approach to this ancient and important problem, opening the door to an `Algebraic Calculus'. Video Content: 00:00 Introduction 4:02 The usual assumptions (of modern texts) 9:16 Area under a parabola was found by Archimedes 18:27 n=2 Back to Archimedes 24:56 Cavalieri's formula 28:07 Serious logical problems with definite integral 34:15 The largely absent 'theory of areas' 38:35 In Linear Algebra we have a different notion of area ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated. Here are the Insights into Mathematics Playlists:    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   Here are the Wild Egg Maths Playlists (some available only to Members!)    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   ************************