Chapter 11

Using the Integral to Solve Problems


Bullet Adding up the area between curves

Bullet Figuring out volumes of weird shapes

Bullet Finding arc length

As I say in Chapter 8, integration is basically adding up small pieces of something to get the total for the whole thing — really small pieces, actually, infinitely small pieces. Thus, the integral


tells you to add up all the little pieces of distance traveled during the 15-second interval from 5 to 20 seconds to get the total distance traveled during that interval. In all problems, the little piece after the integration symbol is always an expression in x (or other variable). For the above integral, for instance, the little piece of distance might be given by, say, math. Then the definite integral


would give you the total distance traveled. Your main challenge in this chapter is simply to come up with the algebraic expression for the ...

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