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Find the mistake in this integration?
The classic derivation of the volume of a cone by the shell method in integration starts with the cone right-side up. And of course there are other ways of finding the volume of a cone by integration. I know, so please do not send me one of these derivations. I just want to see the mistake in the following derivation.
Take an inverted cone, so that
the tip is at the origin,
the maximum radius is the constant R,
the maximum height is the constant H,
h and r stand for the variables of radius and height. Then
Let "Int" = The integral from r= 0 to r = R
Volume= Int 2*pi* r*h dr
Int 2pi*r (H/R) r dr
2pi (H/R Int r^2)dr
2pi (H/R) r^3/3 from 0 to R
2pi*H(R^2)/3
Which is a factor of 2 too much. Where is the error?
Note that it is an INVERTED cone. That means that the first answer here was wrong. At the base of an inverted cone, where the radius = 0, also the height = 0. Someone who read the question please answer.
Also, note that not all integrations find the area between a curve and the axis. Here the integration is "summing up" an infinite number of areas of infinitesimal cylinders, not an area.
Oops, my bad. The first answer was right after all.
1 Answer
- CwCcLv 710 years agoFavorite Answer
it is not true that h = (H/R) r
because at r=0, the expression for the height should give us H, but instead it gives 0. You found the volume of the space under the cone and above the plane of the origin. What you want to use is
h = H - (H/R) r
