Section Summary

In this section we have developed the connections between the processes of antidifferentiation and integration. These connections are described in the two parts of the Fundamental Theorem of Calculus.

First, we found that one way to evaluate a definite integral ${\int }_a^{b}f\left(t\right)dt$ is to find an antiderivative `Ftext[(]t text[)]` of `ftext[(]t text[)]` and then calculate `Ftext[(]b text[)]-Ftext[(]a text[)]`.

Then we introduced the standard name and notation for the long-familiar process of antidifferentiation: the indefinite integral. This name and notation are suggested by the Fundamental Theorem of Calculus, which, as we saw, provides a way to evaluate definite integrals by first finding an antiderivative. We also introduced the evaluation bar notation, which, together with the indefinite integral, allows us to write this aspect of the Fundamental Theorem in a single compact formula:

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

Finally we completed our development of the Fundamental Theorem of Calculus with the observation that the function `F` defined by

is an antiderivative of `f`.

We found that the theorem establishes a very close connection between the apparently rather different processes of definite integration and antidifferentiation. In particular, each can be used to compute the other. It follows that integration (continuous accumulation) and differentiation (instantaneous rate of change) are near-inverse processes.