Chapter Review

We have introduced the second key concept of calculus, that of "continuous accumulation," as expressed in the definite integral,

Roughly speaking, this concept embodies the idea of adding up infinitely many infinitely small quantities. More precisely, we form left-hand sums

whose terms are products of function values and widths of subintervals. The definite integral is then the limiting value of these sums as `n -> oo`.

If the function `ftext[(]t text[)]` is nonnegative for all values of `t` in the interval, then the definite integral represents the area below the graph of `f` and above the interval `[a,b]`. If `vtext[(]t text[)]` is the velocity of an object moving in a straight line, then the integral of `vtext[(]t text[)]` is the distance traveled between the times `t=a` and `t=b`. For any continuous function `f`, the average value of the function over the interval `[a,b]` is

Integration and differentiation are almost inverse operations. The exact relation between the two operations is contained in the Fundamental Theorem of Calculus:

Part I   If `f` is a continuous function from `a` to `b`, then `f` is the derivative of the function `F` defined by

Part II   If `F` is any function whose derivative throughout the interval `[a,b]` is `f,` then

The first part tells us how to construct antiderivatives by definite integration — a powerful idea when using a calculator or computer that can automate integration. The second part tells us how to use antiderivatives to compute definite integrals. We introduced the name indefinite integral of `f` for the family of antiderivatives of `f` and denoted this $\int f\left(t\right)dt$. With this notation and an evaluation bar, we may write the formula in Part II as

`int_a^b ftext[(]t text[)] dt = ``int` `ftext[(]t text[)] dt |_a^b`.