Approximate Integration: Left, Right, Midpoint, and Trapezoidal Rules
In this video I go over the topic of approximate integration and explain how for the situations which the antiderivative can't be found or for experimental data where there is no formula, making an approximation of a definite integral is the only way to go. I go over 4 types of Riemann Sums Approximation methods: Left Endpoint, Right Endpoint, Midpoint and Trapezoidal Approximations methods. I went over Riemann Sums in my earlier video on definite integrals and showed that Riemann Sums are simply approximations of integrals using rectangles but when summed up to infinite they form the exact integral. In this video though, I focus on the 4 already mentioned approximation types and preview the fact that the Midpoint Rule is the most accurate approximation even though the Trapezoidal Rule appears at first glance as the most accurate. I explain this in detail in later videos so stay tuned!
Watch Video On:
- DTube: https://d.tube/#!/v/mes/QmeySL2kDJ49JdRr3pZiAKNnTGBJbYqsHAeXFZ9BF2vvr6
- BitChute: https://www.bitchute.com/video/dxxSTntmowxe/
- YouTube: https://youtu.be/zWzmaLvQU6w
Download Video Notes: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!94835&authkey=!ALofG7kJIusyjGM&ithint=file%2cpdf
View Video Notes Below!
Download These Notes: Link is in Video Description.
View These Notes as an Article: https://steemit.com/@mes
Subscribe via Email: http://mes.fm/subscribe
Donate! :) https://mes.fm/donateReuse 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.
Join my private Discord Chat Room: https://mes.fm/chatroom
Check out my Reddit and Voat Math Forums:
Buy "Where Did The Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
Follow My #FreeEnergy Video Series: https://mes.fm/freeenergy-playlist
Watch my #AntiGravity Video Series: https://mes.fm/antigravity-playlist
- See Part 6 for my Self Appointed PhD and #MESDuality Breakthrough Concept!
Follow My #MESExperiments Video Series: https://mes.fm/experiments-playlist>
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. (#Try2xSpeed)
-- Try 4X+ Speed by Browser Extensions, HookTube.com, Modifying Source Code.
-- Browser Extension Recommendation: https://mes.fm/videospeed-extension- Download and Read Notes.
- Read notes on Steemit #GetOnSteem
- Watch the video in parts.
NOTE #2: If video volume is too low at any part of the video:
- Download this Browser Extension Recommendation: https://mes.fm/volume-extension
Approximate Integration: Left, Right, Midpoint, and Trapezoidal Rules
There are two situations in which it is impossible to find the exact value of a definite integral:
- Sometimes it is very difficult or even impossible to find an antiderivative for some functions
- For example it is impossible to evaluate the following integrals exactly:
- When the function is determined from a scientific experiment through instrument readings or collected data, there may be no formula for the function
- I will go over an example on this in a later video
In both cases we need to find approximate values of definite integrals. We already know one such method:
Recall that the definite integral is defined as a limit of Riemann sums, so any Riemann sum could be used as an approximation to the integrals:
If we divide [a, b] into n subintervals of equal length, then we have:
where:
- xi* = any point in the i-th subinterval [xi-1 , xi].
If xi* is chosen to be the left endpoint of the interval, then:
If f(x) ≥ 0, then the integral represents the area and the left hand approximation, Ln, represents an approximation of the area by the rectangles shown in the above figure.
If we choose xi* to be the right endpoint, then:
The approximations Ln and Rn are called the Left Endpoint Approximation and Right Endpoint Approximation, respectively.
A better approximation is called the Midpoint Rule and is when we choose the xi* to be the midpoint of the subinterval [xi-1 , xi], :
Another approximation, called the Trapezoidal Rule, results from averaging the approximations Ln and Rn:
Although the Trapezoid Rule appears to be the most accurate, actually in most cases the Midpoint Rule is the most accurate! I will compare and explain why in the next video.