In this video I go over another very interesting discovery project, which is a series of advanced math concepts at the end of each chapter in my calculus textbook. This discovery project involves solving the area, surface area, and volume of a shape that is rotated on a slanted line, as opposed to about the typical vertical and horizontal lines which I covered before. I solve question 1 in this video, which involves determining the area of the region with respect to the slanted line. The solution to this video involves using trigonometry to write the terms on the slanted line in terms of x. This is a very interesting video so make sure to watch it and stay tuned for more on this project in my later videos!
Watch Video On:
- DTube:
- BitChute: https://www.bitchute.com/video/jD87hQVTbC6p/
- BitTube: https://bit.tube/play?hash=QmY1fXnvnYkga86mKZeBBEenKruVY3WWdM8qTfhMDqgzph&channel=111547
- YouTube: https://youtu.be/dXlhpDOBDe0
Download PDF Notes: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104297&authkey=!AGZErrrmhurKIX0&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
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.
Discovery Project: Rotating on a Slant: Question 1
We know how to find the volume of a solid of revolution by rotating a region about a horizontal or vertical line.
We also know how to find the surface area of a surface of revolution if we rotate a curve about a horizontal or vertical line.
But what if we rotate about a slanted line?
Let C be the arc of the curve y = f(x) between the points P(p, f(p)) and Q(q, f(q)) and let R be the region bounded by C, by the line y = mx + b (which lies entirely below C), and by the perpendiculars to the line from P and Q.
Question 1
Show that the area of R is:
[Hint: This formula can verified by subtracting areas, but it will be helpful throughout the project to derive it by first approximating the area using rectangles perpendicular to the line as shown above. Use the figure below to help express Δu in terms of Δx.]
