15 Dec '08 05:33

Hello all,

Im trying to derive the formula for volume of a right circular cone fo no particular reason.

hope this is clear,

starting with a right triangle on the x- axis the points are

(a,0);(b,0);(a,z)

the base of the triangle (b-a), and the height (z-0)=z

using the disk method, the radius of the disk is:

z/(a-b)*(x-a) + z,

the equation of a line with a negative slope, passing through the point (a,z), and an x intercept (b,0).

so the volume should equal ( not sure how to format this, see key below)

INT( ) , Integral

a_b , Limits of integration, from a to b

INT(a_b) Pi[z/(a-b)*(x-a) + z]^2 dx

after expanding and evaluating the integral from a to b, I come to:

(Pi*z^2/(a-b)^2)(1/3*b^3 -1/3*a^3 -b^2*a + b*a^2)

but after plugging in some ramdom values for a,b, and z, I come up with 3 times the volume obtained using the formula 1/3*Pi*r^2*h

so I have given up on further simplification for the time being!!

If anyone is willing to put the time in to point out my error I would greatly appriciate it

thanks

Eric

Im trying to derive the formula for volume of a right circular cone fo no particular reason.

hope this is clear,

starting with a right triangle on the x- axis the points are

(a,0);(b,0);(a,z)

the base of the triangle (b-a), and the height (z-0)=z

using the disk method, the radius of the disk is:

z/(a-b)*(x-a) + z,

the equation of a line with a negative slope, passing through the point (a,z), and an x intercept (b,0).

so the volume should equal ( not sure how to format this, see key below)

INT( ) , Integral

a_b , Limits of integration, from a to b

INT(a_b) Pi[z/(a-b)*(x-a) + z]^2 dx

after expanding and evaluating the integral from a to b, I come to:

(Pi*z^2/(a-b)^2)(1/3*b^3 -1/3*a^3 -b^2*a + b*a^2)

but after plugging in some ramdom values for a,b, and z, I come up with 3 times the volume obtained using the formula 1/3*Pi*r^2*h

so I have given up on further simplification for the time being!!

If anyone is willing to put the time in to point out my error I would greatly appriciate it

thanks

Eric