I taught a class today on volumes of solid of revolution; specifically, the ones with holes through them by the so-called “washer method.” For this method you often see the formula
Where R is “outside radius” and r is the “inside radius” and both are functions of x.
It seems to me that this “simplified” form is unduly complicated.
We first worked a problem where a region was rotated around its horizontal edge (Disk method). The region was between , and the line between x = 0 and x = 4, revolved around .
Then I asked them to change the region to that between the graph of and the line again revolved around . These graphs intersect at x = 0 and x = 4. (How convenient!).
Someone immediately had the idea to revolve the line only and subtract the answer from the last answer:
Isn’t that good enough? Is there any need, ever, to set up the washer? Can’t you always subtract the inside volume from the outside volume?
Now I know that
And the latter is shorter and simpler to look at – only one and only one integral sign, but which is really easier to understand and set up? Which shows you really what you’re doing?