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) (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
 = 2\pi r^3)
The volume of a sphere with radius r can be derived many ways using integral calculus. Suppose we have the graph of 
:
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
^2 dx = \pi (r^2 - x^2) dx)
Integrating from -r to r,
\,dx)
(evaluate at x = r and subtract the value obtained at x = -r)
 - \frac{r^3}{3}) - (r^2(-r) - \frac{-r^3}{3}))
Comparing this with our expression for the volume of the cylinder, we see that
and we are done.
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