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Cylindrical Slice (Posted on 2005-11-25) Difficulty: 3 of 5
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.

See The Solution Submitted by Brian Smith    
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Solution single integral | Comment 2 of 13 |

Taking each element as a rectangular vertical slice of height h*x/r and width 2*(r^2-x^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^2-x^2) dx

With the help of Wolfram's The Integrator, this integrates to

[- (2h/(3r)) (r^2-x^2)^(3/2)]{0 to r}

which equals 2h/(3r) r^3 = 2 h r^2 / 3

  Posted by Charlie on 2005-11-25 11:56:00
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