In this video I go over further into Polar Coordinates and this derive the formula for determining the area of a polar curve. In doing this I make note of the formula for the area of a sector of a circle which I covered in my earlier video, and is equal to ½r2ϴ. Then just like in deriving the formula for the area under a curve in basic Cartesian coordinates, we divide the region into many parts, but the only difference now is that split the angle into many subintervals of equal angle Δϴ. Now instead of summing up an infinite amount of infinitely small rectangles, we instead sum up sectors of a circle. Then we can write the summation as a limit as the intervals approach infinity, which when we consider it as a Riemann Sum we get the formula for the area of the polar curve. Note though that the summation I derived is not exactly as the Riemann Sum formulas I derived in my earlier videos, because now we are dealing with a squared function, this is in fact still the case and which I may prove in a later video so stay tuned! This is a very useful video to understand how to derive the formula for the area of a polar curve which I will be utilizing in my later videos, and also a good way to compare with the Cartesian area derivation, so make sure to watch this video!
Watch video on:
Download video notes: https://1drv.ms/b/s!As32ynv0LoaIhvUPWn8q0ZxQ2cqz_A
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/matheasy
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
Join 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
In this section we develop the formula for the area of a region whose boundary is given by a polar equation.
We need to use the formula for the area of a sector of a circle (see my earlier video):
Where r is the radius and θ is the radian measure (i.e. angle in radians) of the central angle.
Note that the area of a sector of a circle is proportional to its central angle relative to a circle's central angle, 2π.
Let R be the region, shown below, bounded by the polar curve r = f(θ) and by the rays θ = a and θ = b, where f is a positive continuous function and where 0 < b - a ≤ 2π.
We divide the interval [a , b] into subintervals with endpoints θ0 , θ1 , θ2 , … , θn and equal width Δθ.
The rays θ = θi then divide R into n smaller regions with area Ai of the i-th region is approximated by the area of the sector of a circle with central angle Δθ and radius f(θi*).
And so an approximation to the total area A of R is:
It appears from the above figure that the approximation improves as n → ∞.
Thus taking the limit and writing it as a Riemann sum we get the following:
The only issue with this is that the Riemann sum above is for the function g(θ) = 1/2[f(θ)]2 as opposed to the non-squared version f(θ) which I covered in my earlier videos.
But it is nonetheless plausible (and can in fact be proved, which I may do in a later video) the formula for the area A of the polar region R is in fact:
Note the similarity between the formula of a sector of a circle and that of a polar curve.
This is because the area of a polar curve is derived from the area of a sector of a circle!
When we apply the above formula, it is helpful to think of the area as being swept out by a rotating ray through O that starts with angle a and ends with angle b.
