DEFINITION Regular Partition Suppose [a, b]is a closed interval containing n subintervals x of equal length△x= with a=xo and b=x. The endpoints xo, xi, x2 In-I, xn of the subintervals are called grid points and they create a regular partition of the interval [a, b]. In general, the kth grid point is xk=a+ kAx for k =0, 1, 2,...,n Copyright 2011 Pearson Education, Inc. Publishing as Pears on Addis on-Wesley Slide 5-12
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Slide 5 - 12
Area of kth rectangle = height·base =fGx)△x FIGURE 5.8 Copyright 2011 Pearson Education, Inc. Publishing as Pears on Addis on-Wesley Slide 5-13
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Slide 5 - 13
DEFINITION Riemann Sum Suppose f is defined on a closed interval [a, b], which is divided into n subintervals of equal length A r If xk is any point in the kth subinterval[xk-1, xk],then f(x1)Δx+f(x2)△x+…+f(xn) is called a Riemann sum for f on [a, b]. This sum is a left Riemann sum if xx is the left endpoint of [xk-I, xk( Figure 5.9); a right Rie mann sum if xk is the right end point of [xk-I,xk(Figure 5.10); and a midpoint Riemann sum if xx is the midpoint of [xk-I,xk(Figure 5. 11), for k=1,2,.,n Copyright 2011 Pearson Education, Inc. Publishing as Pears on Addis on-Wesley Slide 5-14
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Slide 5 - 14
Left riemann sum ETO FIGURE 5.9 Copyright 2011 Pearson Education, Inc. Publishing as Pears on Addis on-Wesley Slide 5-15
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Slide 5 - 15
Right riemann sum O x=b FIGURE 5.10 Copyright 2011 Pearson Education, Inc. Publishing as Pears on Addis on-Wesley Slide 5-16
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley. Slide 5 - 16