Question

I'm trying to understand how Cavalieri's Principle is used in the "Napkin Ring Problem"
Can someone help me?

Answer 1

I've already read the wikipedia article

Answer 2

I love seeing a post with two mods. I smell the competition @hartnn @ganeshie8 ;) http://prntscr.com/ael8k0

Answer 3

I get that through a simple computation, the area of the cross-section is the same as that of the corresponding cross-section of a sphere of radius h/2.
(Assuming that the diameter of the cylinder is h)

Answer 4

Is there some way to understand this intuitively rather than using equations, only?

Answer 5

Iʻve never had problems with my napkin, but when I do, Iʻll be sure to come back and refer to this thread.

Answer 6

competition with ganesh!? NO WAY!
here I am removing a napkin from my pocket and making a ring :P

Answer 7

if i understand correctly, you're trying to find the volume of a sphere with a hole through the diameter :

http://i.ebayimg.com/images/a/%28KGrHqR,!iQFCT7!eUjGBQoGuDjtj!~~/s-l1600.jpg

Answer 8

wait. actually, I just understood it!
sorry for all the trouble.
(sometimes, my brain won't just work the way I want it to...)

Answer 9

Nice! Have you worked it using just the Cavalieri's principle, w/o any calculus ?

Answer 10

yep!
very elegant, in my opinion

Answer 11

May I see your work ?

Answer 12

give me a sec

Answer 13

sure... this is interesting... i don't see any cross sections having the same area in the napkin ring or the hole hmm

Answer 14

Sorry it took while.
messed up my drawing

Answer 15

aren't you using cylindrical shells ?

Answer 16

i still don't see how your solution is related to Cavalieri's principle

Answer 17

Left above diagram shows the volume that we have to find, which is shaded in blue

Answer 18

At the right, you can see that I've shaded the cross section of the volume that we need to find with light green

Answer 19

If you see this from above, it would look like the purple circle with a green-circle shaped hole in the middle

Answer 20

that area is equal to the cross section of a sphere with radius of h/2

Answer 21

yeah like a washer

Answer 22

cavalieri's principle states that if ever plane intersects both regions in cross-sections of equal area, then the 2 regions have equal volumes.

Answer 23

using that, since the cross sections are the same, the volume that we're trying to find is the same as the volume of a sphere with radius of h/2, which is \[\pi h^{3}/6\]

Answer 24

Has anybody felt the pain of doing a washer with a dy?

Answer 25

are you saying the area of washer is equal to the corresponding cross section of the sphere of radius h/2 ?

Answer 26

yes

Answer 27

awesome! i get it now, truely beautiful !

Answer 28

well, i guess than everything is good, now

Answer 29

i guess that explanation is the best way to learn :)