The method's formula can be derived as follows:
Consider the function which describes our cross-section of the solid, now the integral of the function can be described as a Riemann integral:
The Riemann sum can be thought up as a sum of a number n of rectangles with ever shrinking bases, we might choose one of them:
Now, when we rotate the function around the axis of revolution, it is equivalent to rotating all of these rectangles around said axis, these rectangles end up becoming a hollow cylinder, composed by the difference of two normal cylinders. For our chosen rectangle, its made by obtaining a cylinder of radius with height , and substracting it another smaller cylinder of radius , with the same height of , this difference of cylinder volumes is:
By difference of squares , the last factor can be reduced as:
The third factor can be factored out by two, ending up as:
This same thing happens with all terms, so our total sum becomes:
In the limit of , we can clearly identify that:
- as goes to 0 ends up becoming
- becomes itself, going from a to b (ignoring the last term which is now infinitesimal)
- becomes an infinitesimal
Thus, at the limit of infinity, the sum becomes the integral:
QED .
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