1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 5: The Integral Section 5.1: Approximating and Computing Area Jon Rogawski Calculus, ET First Edition

2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Example, Page 308 2. Figure 14 shows the velocity of an object over a 3-min interval. Determine the distance traveled over the intervals [0, 3] and [1, 2.5]. Remember to convert mph to mi/min. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Combining Figures 2 and 3, we may construct N intervals of equal width, Δx = (b – a)/N, over the interval [a, b] such that the right endpoints of the intervals are given by:

7. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Since we used rectangles of common width, the approximate area under the curve becomes: Using the second rule of linearity of summations, we obtain Some texts refer to this method as the Rectangular Approximation Method or RAM.

8. Example, Page 308 6. Use the following table of values to estimate the area under the graph of f (x) over [0, 1] by computing the average of R 5 and L 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company So far, we have considered only approximations using the value of f (x) on the right of the rectangle, but we may also use the value at the left or the midpoint, as shown below.

10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The summations we use for the left-hand and midpoint approximations are respectively: The Rectangular Approximation Methods may further be abbreviated as RRAM, LRAM, and MRAM to indicate using the y-values on the right, left, and midpoint respectively.

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figures 7 and 8 illustrate using midpoint, left and right approximations of the area under a parabola and also, the amount of error in using each method.

12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figures 8 and 9, we observe using RRAM and LRAM to approximate the area under curves that are either continuously increasing or decreasing. Such curves are said to be monotonic. Notice RRAM underestimates areas for functions that are monotonic decreasing and over- estimates those mono- tonic increasing. LRAM does the opposite.

13. Example, Page 308 10. Estimate R 2, M 3, and L 6 for the graph in Figure 16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The benefit of using more rectangles illustrated in Figure 10 leads to Theorem 1:

15. Example, Page 308 10. Estimate R 2, M 3, and L 6 for the graph in Figure 16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Homework Homework Assignment #34 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company