A right cylinder has height h and radius r. It is sliced by a plane that is tangent to one circular base at A and intersects the other at diameter BC. What is the volume of slice ABCD?
Note that BO=CO=DO=r, AD=h, BC is perpendicular to DO, and AD is perpendicular to DO.
Taking each element as a rectangular vertical slice of height h*x/r and width 2*(r^2x^2)^(1/2), we can integrate the product of these from x=0 to r.
Integ{0 to r} h (x/r) 2 sqrt(r^2x^2) dx
With the help of Wolfram's The Integrator, this integrates to
[ (2h/(3r)) (r^2x^2)^(3/2)]{0 to r}
which equals 2h/(3r) r^3 = 2 h r^2 / 3

Posted by Charlie
on 20051125 11:56:00 