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!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvUPWn8q0ZxQ2cqz_A
View video notes on the Hive blockchain: https://peakd.com/@mes/video-notes-polar-coordinates-area-formula
Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Area of a Sector of a Circle: Proof: https://youtu.be/lglnySF8DyY
Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I
Integration Overview: How are Riemann Sums, Antiderivatives, and Integrals Linked?: https://youtu.be/TGnnu1vnD_U .
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