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

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

In this continuation of our Famous Math Problem 10, we derive Cavalieri's quadrature formula for polynomials in a new, completely algebraic way, avoiding any use of `real numbers', `infinite processes' and `limits'. The key tool is rather the Binomial theorem, and five basic properties that any reasonable theory of area/integration should satisfy: Linearity, Additivity, Translation Invariance, Scale Invariance and Normalization. Along the way we give brief overviews of different classical approaches to setting up integration theory, such as the Riemann Integral, the Lebesgue Integral and some others. One should be highly suspicious of these theories, which, contrary to popular opinion, have NOT been developed logically and clearly. This is why an algebraic alternative such as we initiate here is so important. Video Content: 00:00 Introduction 1:54 Possible approaches: Riemann Integral 5:49 Lebesgue Integral (simplified view) 08:23 Piecewise linear integral and Polygonal approximation 10:41 Properties of an integral 14:35 Scale Invariance 18:40 Fundamental Integral formula and proof 33:50 Consequences ************************ 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   ************************