Chapter 6 is applications of integration

There's a big trick to integrals in these volume and work problems. When we break things into simple shapes, like disks or rectangular slabs (or triangular slabs, or trapezoidal slabs), we always cheat in the dx direction. That slab has short sides that may be curved or slanted. No matter, as long as you're cheating on the slab's shape on the one side that you're going to take as small as possible in the calculation. The beauty of the calculus is that that acceptable error actually disappears entirely. We're actually finding the exact volume/work when we pass to the limit, as dx ---> 0. In other words, as dx vanishes, so does what little error our shabby constructions introduce! Cool thing about the integral sign and the dx is that it's saying "Pass to the limit" every time you write it. The key is that the function between those two symbols is continuous (although much of higher calculus is all about just how much discontinuity you can tolerate and how to deal with it).

Perfect for a guy like me who can't walk a straight line, let alone sketch a parabola that's at all true to scale... I'll never be a carpenter, but maybe I'll be able to help one, one day, and get some free cabinetry outta the deal.

6.1 - Area between two functions is the integral of their positive difference (Big function on top.)
6.2 - Volume of solids of revolution - The Disk method, where the representative volume is a circular disk of thickness dx (or dy)
6.3 - Volume of solids of revolution - The Shell method, where the representative volume is the sidewall of a soup can of thickness dx (or dy)
6.4 - WORK - Work is force times distance, with each being continuous functions of position x.
6.5 - Mean Value Theorem - There's at least one location on the graph whose height is the average value of the continuous function, and we use an integral to find the average height.

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