“George, I really like doing things backward. Can we do differentiation backward?”
“Sure we can, Lenny. It’s called integration.”
Differential calculus has to do with rates of change. Integral calculus has to do with sums of many tiny incremental quantities. It’s not immediately obvious that these have anything to do with each other, but they do.
We begin with the graph of a function f(t), as in Figure 1.
Figure 1: The behavior of f(t).
The central problem of integral calculus is to calculate the area under the curve defined by f(t). To make the problem well defined, we consider the function between ...