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?
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?
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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
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