In this video, I recap on the important concept of summing up rectangles below a curve and illustrate how the definite integral is basically defined to express the limit of the area of rectangles under a curve. I also provide a brief history lesson on Bernhard Reimann, who first introduced the summation of the rectangles idea which we adopt for the definition of the definite integral.
Watch video on:
- 3Speak:
- Odysee: https://odysee.com/@mes:8/the-definite-integral-and-riemann-sums:6
- BitChute:
- Rumble:
- DTube:
- YouTube:
Download video notes: https://1drv.ms/b/s!As32ynv0LoaIiZxXBE0C5IkIE0gJzg?e=BWKPsE
View Video Notes Below!
Download these notes: Link is in video description.
View these notes as an article: @mes
Subscribe via email: http://mes.fm/subscribe
Donate! :) https://mes.fm/donate
Buy MES merchandise! https://mes.fm/store
More links: https://linktr.ee/matheasyReuse of my videos:
- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.
Fight back against censorship:
- Bookmark sites/channels/accounts and check periodically
- Remember to always archive website pages in case they get deleted/changed.
Recommended Books:
- "Where Did the Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
Subscribe to MES Truth: https://mes.fm/truth
MES Links Telegram channel: https://t.me/meslinksJoin my forums!
- Hive community: created/hive-128780
- Reddit: https://reddit.com/r/AMAZINGMathStuff
- Discord: https://mes.fm/chatroom
Follow along my epic video series:
- #MESScience: https://mes.fm/science-playlist
- #MESExperiments: @mes/list
- #AntiGravity: @mes/series
-- See Part 6 for my Self Appointed PhD and #MESDuality breakthrough concept!- #FreeEnergy: https://mes.fm/freeenergy-playlist
- #PG (YouTube-deleted series): @mes/videos
NOTE #1: If you don't have time to watch this whole video:
- Skip to the end for Summary and Conclusions (if available)
- Play this video at a faster speed.
-- TOP SECRET LIFE HACK: Your brain gets used to faster speed!
-- MES tutorial: @mes/play-videos-at-faster-or-slower-speeds-on-any-website- Download and read video notes.
- Read notes on the Hive blockchain #Hive
- Watch the video in parts.
-- Timestamps of all parts are in the description.Browser extension recommendations:
- Increase video speed: https://mes.fm/videospeed-extension
- Increase video audio: https://mes.fm/volume-extension
- Text to speech: https://mes.fm/speech-extension
--Android app: https://mes.fm/speech-android
The Definite Integral and Riemann Sums: Introduction
The Definite Integral
- Showed the limit above arises for Area and Distance
- It turns out that this same type of limit occurs in many situations:
- Ex. Lengths of curves, volumes of solids, centers of mass, force due to water pressure, work, etc.
- Thus this limit deserves a special name and notation
Definition of a Definite Integral
- The symbol ∫ was introduced by Leibniz and is called an integral sign.
- It is an elongated S and was chosen because an integral is a limit of sums
- f(x) is called the integrand
- a, and b are called the limits of integration
- a is the lower limit
- b is the upper limit
- The symbol dx has no official meaning by itself
- ∫ab f(x)dx is all one symbol
- The procedure for calculating an integral is called integration
- The sum ∑i=1n f(xi* )∆x is called a Riemann Sum after the German mathematician Bernhard Riemann (1826-1866).
- History Lesson: Bernhard Riemann received his Ph.D under the direction of the legendary Gauss at the University of Gottingen and remained there to teach. The definition (above) of an integral that we use is due to Riemann. He also made major contributions to the theory of functions of a complex variable, mathematical physics, number theory, and the foundations of geometry. Riemann’s broad concept of space and geometry turned out to be the right setting, 50 years later, for Einstein’s general relativity theory. Riemann’s health was poor throughout his life, and he died of tuberculosis at the age of 39.
Example 1
Solution: