1. Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A function is integrable over [a, b] if all of the Riemann sums (not just the endpoint and midpoint approximations) approach one and the same limit L as the norm of the partition tends to zero, which we may write as:. Assuming |R(f, P, C) – L| gets arbitrarily small as the norm||P|| tends to zero, the limit is called the definite integral of f (x) over [a, b].

4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The definite integral is usually referred to as the integral of f over [a, b]. The function f (x) inside the integral symbol is called the integrand. The numbers a and b are called the limits of integration. The independent variable in the function is used as the variable of integration.

5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 3, when f (x) is not positive for all x on [a, b], the definite integral yields the signed area between the graph and the x-axis. Signed area is defined as: If the graph in Figure 3 represented the velocity of a particle, then the integral from a to b would tell us the net displacement or movement of the particle from its starting point.

6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company For the function illustrated in Figure 4.B. the Riemann sum converges to the signed area. In summary,

7. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Calculate the definite integral of f (x) = 2x – 5 over [0, 3].

8. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Evaluate the definite integral of f (x) = 3 – x over [0, 4].

9. Example, Page 321 Use the basic properties of the integral and the formulas in the summary to calculate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 8 illustrates the definite integral of f (x) = C over [a, b] for some C > 0. The integral may be evaluated by using Theorem 2.

11. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Similar to limits, definite integrals have linearity properties as noted in Theorem 3.

12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we reverse the limits of integration, we change the sign of the signed area yielded by the definite integral as noted in the following definition: If the upper and lower limits of integration equal one another, the width of the interval is zero and

13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Show that for all values of b, (positive and negative):

14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 9 more clearly illustrates the results obtained on the previous slide.

15. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company This theorem is illustrated in Figure 10.

16. Example, Page 321 Calculate the integral, assuming Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 11 illustrates Theorem 5, the Comparison Theorem.

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

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If a function has a lower bound m and an upper bound M on [a, b], then the Comparison Theorem may be written algebraically as: This is illustrated in Figure 12.

20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Using the information given in Figure 13, find the bounds of the definite integral of x –1 on [0.5, 2].

21. Example, Page 321 Calculate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

22. Homework Homework Assignment #2 Read Section 5.3 Page 321, Exercises: 1 – 61(EOO), 65, 71 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company