In this video I go over further into improper integrals and this time discuss the type 1, Infinite Intervals, in great detail as well as outlining a definition for it. The first type of improper integrals is when the interval is infinite, such as the integral of a function from x = a to infinity. Even though the interval may be infinite the actual integral may approach a finite number if the limit of the function at positive or negative infinity exists.
Watch Video On:
Download PDF Notes: http://1drv.ms/1L0BkCr
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
Follow My #MESExperiments Video Series: https://mes.fm/experiments-playlist>
NOTE: 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.
Improper Integrals: Type 1: Infinite Intervals
In my earlier video I went over an Introduction to Improper Integrals and briefly discussed the two types:
Type 1: Infinite Intervals
Type 2: Infinite Discontinuity (or Discontinuous Integrands)
In this video I will go over the Type 1 improper integrals in greater detail as well as outlining the definition.
Type 1: Infinite Intervals
Consider the infinite region S that lies below the curve y = 1/x2, above the x-axis, and to the right of the line x = 1.
You might think that, since S is infinite in extent, its area must be infinite, but let's take a closer look.
The area of the part of S that lies to the left of the line x = t is:
Notice that A(t) < 1 no matter how large t is chosen to be.
We also observe that:
The area of the shaded region approaches 1 as t → ∞, so we say that the area of the infinite region S is equal to 1 and we write:
Using this example as a guide, we define the integral of f (not necessarily a positive function) over an infinite interval as the limit of integrals over finite intervals.
Definition of an Improper Integral of Type 1
In part (c) any real number a can be used.
Any of the improper integrals in the above Definition can be interpreted as an area provided f is a positive function.
