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Theorem 3: The Addition Property

THEOREM 3 (The Addition Property)

Suppose f is continuous on an interval I. Then for all a, b, c in I,


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


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



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 integral04_integration-116.gifis a positive number and the

integral04_integration-117.gifis 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 04_integration-119.gif 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 04_integration-120.gifis 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

Last Update: 2006-11-05