Chapter 6 The Definite Integral

There are two fundamental problems of calculus 1.Finding the slope of a curve at a point 2.Finding the area of a region under a curve Our study of the derivative has addressed 1. In chapter 6, we will develop techniques for addressing 2.

6.1 Antidifferentiation

We have developed techniques for calculating the derivative F’(x) of a function F(x). In many applications, it is necessary to proceed in reverse. Given a derivative F’(x), we must find the function F(x). The process of determining F(x) from F’(x) is called antidifferentiation.

Suppose f(x) is a given function and F(x) is a function having f(x) as its derivative, then F’(x) = f(x). We call F(x) the antiderivative of f(x).

Problem: Find the antiderivative of f(x) = x 2. What is the function whose derivative is x 2 ? Consider what must happen in order to have a derivative that is x 2. x 2 = kx k-1, by the power rule. then k - 1 must be 2, so k = 3. But the derivative we have is x 2, not 3x 2. What happened to the leading 3?

The three must have been multiplied by something that made it result in there being a leading 1 instead of a 3. The antiderivative could have been Note that

Is the only solution/antiderivative? What about

The observation that there are multiple antiderivatives of f(x) leads us to the following theorem. Theorem I If F 1 (x) and F 2 (x) are two antiderivatives of the same function f(x), then F 1 (x) and F 2 (x) differ by a constant. In other words, there is a constant C such that F 2 (x) = F 1 (x) + C

Geometrically, the graph of F 2 (x) is obtained by shifting the graph of F 1 (x) vertically.

An associated theorem illustrates the following fact Theorem II If F’(x) = 0 for all x, then F(x) = C for some constant.

Using Theorem I, we can find all antiderivatives of a given function once we know one antiderivative. For example, since one derivative of x 2 is (1/3)x 3, all antiderivatives of x 2 have the form (1/3)x 3 + C where C is a constant.

Suppose that f(x) is a function whose antiderivatives are F(x) + C. The standard notation to express this fact is The symbol is called an integral sign, and the entire notation is called an indefinite integral and stands for the antidifferentiation of the function f(x). We indicate the variable of interest by following f(x) with the variable prefaced by dx.

Rules for Antidifferentiation