What's the length of the curve f(x) = x2 on the interval [1,2]?
The part of the curve we're interested in is
We chop the curve up into little bits. The length of each little bit is approximately
since f '(x) = 2x.
We're looking at the interval [1,2], so we need an integral from 1 to 2 to get the exact length of the curve:
That's it. Unlike areas and volumes, we don't need to make bunnies, coffee cups, or other strange oddities from the information about the geometries we start with.
Because these problems are so simple, we can skip the step where we write down the length of a little bit of the curve and just go directly to the integral if we want. In the example, it would have been ok for us to just write
"a = 1, b = 2, and f '(x) = 2x. So the length of the curve is ."