Theorem 3: The Addition Property

This property is illustrated in Figure 4.2.5 for the case a < b < c. The Addition Property holds even if the points a, b, c are in some other order on the real line, such as c < a < b.

Figure 4.2.5

PROOF

First suppose that a < b < c. Choose a dx that evenly divides the first interval length b-a. This simplifies our computation because it makes b a partition point, b = a + H dx. Then, as Figure 4.2.6 suggests,

Taking standard parts we have the desired formula

Figure 4.2.6

To illustrate the other cases, we prove the Addition Property when c < a < b. The previous case gives

Since reversing the endpoints changes the sign of the integral,

and the desired formula

follows.

The definite integral of a curve can be thought of as area even if the curve crosses the x-axis. The curve in Figure 4.2.7 is positive from a to b and negative from b to c, crossing the x-axis at b. The integralis a positive number and the

integralis a negative number. By the Addition Property, the integral

is equal to the area from a to b minus the area from b to c. The definite integral always gives the net area between the x-axis and the curve, counting areas above the x-axis as positive and areas below the x-axis as negative.

The definite integral is a real function of two variables u and v

and does not depend on the dummy variable t. If we replace u by a constant a and v by the variable x, we obtain a real function of one variable x, given by

Our fourth theorem states that this new function is continuous.

Figure 4.2.7