This week's column in the series Math: From Basic To Baffling deals with integral calculus: It Slices, It Dices. I find it unsatisfying -- I think it's because it lacks a fully solved demo calculation (like many of his previous columns do). There's a lot of tell, but very little show.
[Interestingly, some of the show is left to the footnotes and citations to books and articles, but much of it is not accessible online -- I would think that this is a good reason to use and explain some of that material in the column itself. Having said that, one of his footnotes does take us to this really cool demo -- or this one -- that relates the volume of a sphere, a cone and a cylinder. Mamikon Mnatsakanian of Caltech has many more of such goodies.]
Anyways, here's an excerpt:
In a 3-D generalization of this method, Archimedes (and before him, Eudoxus, around 400 B.C.) calculated the volumes of spheres, cones, barrels, prisms and various other solid shapes by re-imagining them as stacks of many wafers or discs, like a salami sliced thin. By computing the volumes of the varying slices, and then ingeniously integrating them — adding them back together — they were able to deduce the volume of the original whole.
Today we still ask budding mathematicians and scientists to sharpen their skills at integration by applying them to these classic geometry problems. They’re some of the hardest exercises we assign, and a lot of students hate them, but there’s no surer way to hone the facility with integrals needed for advanced work in every quantitative discipline from physics to finance.