SOLUTION: Archimedes showed that volume of a sphere is two-thirds the volume of the smallest right circular cylinder that can contain it. Verify this.

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Question 455027: Archimedes showed that volume of a sphere is two-thirds the volume of the smallest right circular cylinder that can contain it. Verify this.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The smallest right circular cylinder that will contain the sphere (radius = r) is one with a radius of r and a height of 2r. The volume of such a cylinder is



The volume of a sphere with radius r can be derived many ways using integral calculus. Suppose we have the graph of :

graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+sqrt%2864-x%5E2%29%29

Using the solids of revolution technique (http://en.wikipedia.org/wiki/Solid_of_revolution), if we rotate the graph about the x-axis to produce a sphere, we can take a differential part (dx), and for each dx, the volume of the respective cylinder is . However, y is a function of x, so if we replace y with , we get



Integrating from -r to r,



(evaluate at x = r and subtract the value obtained at x = -r)





Comparing this with our expression for the volume of the cylinder, we see that



and we are done.