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Unformatted text preview: (3/20/08) Section 6.2 The definite integral Overview: We saw in Section 6.1 how the change of a continuous function over an interval can be calculated from its rate of change if the rate of change is a step function. We also outlined there how in other cases, changes in the function might be determined as limits by using step function approximations of the rates of change. In this section we use this idea in the definition of the definite integral . Then we derive some of the basic properties of the integral. Topics: The definite integral Piecewise continuous functions Integrals and areas Special Riemann sums Properties of definite integrals Unbounded functions Riemann sum programs The definite integral The definite integral of the function y = f ( x ) from x = a to x = b with a < b is a number, denoted integraldisplay b a f ( x ) dx . The symbol integraltext is called an integral sign , the numbers a and b are the limits of integration , [ a,b ] is the interval of integration , and f ( x ) is the integrand . As we explained at the end of the last section, we want to define the integral so that for the function f of Figure 1, the integral from x = a to x = b equals the area of region A between the graph and the x-axis where f ( x ) is positive, minus the area of region B between the graph and the x-axis where f ( x ) is negative. To accomplish this, we approximate y = f ( x ) by step functions whose graphs form approximations of the two regions by rectangles, as in Figure 2. The integral is defined to be the limit, as the number of rectangles tends to and their widths tend to zero, of the area of the rectangles above the x-axis, minus the area of the rectangles below the x-axis. x y y = f ( x ) a b c A B t v y = f ( x ) a b FIGURE 1 FIGURE 2 180 Section 6.2, The definite integral p. 181 (3/20/08) To construct approximating rectangles, we start with a partition a = x < x 1 < x 2 < < x N- 1 < x N = b (1) of the interval [ a,b ]. It divides [ a,b ] into N subintervals, [ x ,x 1 ] , [ x 1 ,x 2 ] ,..., [ x N- 1 ,x N ]. The j th subinterval is [ x j- 1 ,x j ] for j = 1 , 2 , 3 ,...,N (Figure 3). We let x j denote its width: [The width of the j th subinterval] = x j = x j- x j- 1 . We also pick, for each j , a point c j in the j th subinterval that is in the domain of f . x x j- 1 x j x j c j The j th subinterval [ x j- 1 ,x j ] FIGURE 3 Figure 4 shows the graph of the function f of Figures 1 and 2, and five rectangles that correspond to a partition a = x < x 1 < x 2 < x 3 < x 4 < x 5 = b of [ a,b ] into five subintervals and to points c 1 ,c 2 ,c 3 ,c 4 and c 5 in the subintervals. For j = 1 and 2 on the left, f ( c j ) is positive, the base of the rectangle is the j th subinterval on the x-axis, and its top is at y = f ( c j ). For j = 3 , 4 , and 5 on the right, f ( c j ) is negative, the top of the rectangle is the j th subinterval on the x-axis and its base is at y = f ( c j )....
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