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Home Integral Theorems of Calculus Theorem 3: The Addition Property  
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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 PROOF First suppose that a < b < c. Choose a dx that evenly divides the first interval length ba. 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 xaxis. The curve in Figure 4.2.7 is positive from a to b and negative from b to c, crossing the xaxis 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 xaxis and the curve, counting areas above the xaxis as positive and areas below the xaxis 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


Home Integral Theorems of Calculus Theorem 3: The Addition Property 