Theorem of Pappus (KristaKingMath)
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- เผยแพร่เมื่อ 17 ม.ค. 2025
- ► My Applications of Integrals course: www.kristaking...
Learn how to use the Theorem of Pappus to find the volume of a solid, in this particular case, a right circular cone. Theorem of Pappus tells us that volume is equal to area of the plane region, times the distance traveled by the centroid of the same plane region, if the plane region is revolved around the x-axis.
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Thanks for the response! It's quite amazing that the centroid has this property. I may be overthinking it, but we are taking the product of a one and two dimensional object and getting a three dimensional result. Very cool!
Think about the plane region, basically, a 2D triangle that sits inside the cone. The centroid of that triangle (basically the middle of it), will travel a distance when you revolve that triangle around the y-axis. That center dot will essentially form a circle when you rotate the region, and the circumference of that circle is the distance traveled by the centroid. I hope that makes sense!! :)
You managed to do what my profesor couldn't in a full lesson. Thanks for clearing up some doubts that I had.
A segment of a circle of radius R is bounded by an arc equal to 1/3 the circumference of the circle. Find the volume of the solid generated by revolving this segment about its chord as an axis. (This solid is know as Kepler's Lemon). Note: Do note use integral calculus.
this video is not just helpful . it's helpful , great and awesome . actually you're doing great job . thanks for that inexpressible effort ..
+ahmed saeed You're welcome, I'm happy to help!
I didn't see VeryEvilPettingZoo's response before I responded. Thanks for helping each other... that's awesome!! :D
Glad I could help! :)
Thanks! :)
Best teacher ever
Thank you so much!!
Props young lady. Helped me with my Calc 2 homework, thanks!
You're welcome, I'm so glad it helped! :)
What exactly does "distance traveled by the centroid of the plane region" mean; from where to where? Thanks for the great videos!
thanks!! :D
very WELL DONE, KRISTA!!.. You RULE.. :) ... by the way... a Great CHECK to see if your result is Correct.. (not that I think it wasn't correct!! ) .. is to Note that the Volume of a Cone is 1/3 the Volume of a CYLINDER using the same values for "h" and "r" .. and SURE ENOUGH.. you arrived at the 1/3(Pi)(r^2)(h) !!!! BINGO!! :) ... the area of the cylinder would be (Pi)(r^2)(h) !! THANK YOU KRISTA for all your wonderful AND very Educational Videos!! ..
could you not also find the equation of the line and rotate it around the y-axis by using it's bounds and the disk method? not positive if that would work but i feel like it could be a substitute
Well done.
Thanks .
so simple! thank you!
thanks, never understood any of what my professor explained for 2 hours. but in this video it took only around 10 mins
So glad I could help simplify it for you! :)
Yes L, Krista's videos give us "working knowledge"; a whole new generation of engineers is on the way, imho.
your method has helped me is there any method to analyze 3D problems.
Thankyou
when do we need the y coordinate..does the distance formula also change in that case?
hey Krista I want to find center of mass of a rigid uniform body like that semi ring please help me out
from where did u get the board and chalk on ur computer
I explain here :) www.kristakingmath.com/my-videos
Where does the 1/A come from. I don't get it. Can anyone explain?
when do you use this theorem on real life?
hello
can you state this theorem
i mean statement of this theorem
I just had one doubt. Can you explain why did we consider only the x-coordinate and not y-coordinate?
+Aman Sharma well if u are trying to find the distance of a straight line (which is in this case)u can neglect ur y corrdinate
Cos The triangle so form is rotating about y-axis,
If it's rotating about x-axis, then we'll consider only y coordinate and not x centroid
thank you
You're welcome! :)
Thank you very much! :)
You're welcome!
I don't understand why the equation of F(x) is taken as the hypotenuse of the right angled triangle.
Bcos it is the line that enclosed the cone and its equation is y=f(x)
If you're desi like me you can't stop giggling at "Pappu's theorem"
video is great
but still have question about Y' is the formula of Y' is correct
X' is verified from my side
please help me with this matter
I love this theorem!!!
can somebody recap simply the problem for me plz ? it's for a project about Descartes for school...
I don't understand why you can't just multiply 1/2 b (h)(2pi)(r)^2
inlove with you
WTF is going on
thanks a lot, but there was a mistake in your solution. after the integration, r cubed was changed to r squared. and with that being done, your result is wrong.
Yaarub Alanie it looks like the r's just canceled out. 1/3^r Xr^3
oh ok. i am sorry.
thanks again
I AM SO COOKED FROM FINALS THAT I tried and pick up my mouse and dial it.