1. Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 398 Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 398 Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 398

11. Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 398 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

14. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

15. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

16. Homework, Page 398 Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

17. Homework, Page 398 49.Sketch the hypocycloid x 2/3 + y 2/3 = 1 and find the volume of the solid obtained by revolving it about the x-axis.

18. Homework, Page 398 51.A bead is formed by removing a cylinder of radius r from the center of a sphere of radius R. (Figure 12) Find the volume of the bead with r = 1 and R = 2.

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 7: Techniques of Integration Section 7.1: Numerical Integration Jon Rogawski Calculus, ET First Edition

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

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The shaded area in Figure 1 cannot be calculated directly using a definite integral, since there is not an explicit antiderivative for Instead, we will rely on numerical approximation using the trapezoidal method

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we divide the interval [a, b] into N even intervals, the area may be found using the Trapezoidal Rule

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As shown in Figure 3, the area of the trapezoidal segment is equal to the average of the left- and right- RAM areas. As shown in table one, by increasing the size of N, we can attain whatever degree of accuracy we may need.

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

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 5 illustrates how a mid point estimate rectangle has the same area as a trapezoid where the top of the trapezoid is tangent to the curve at the midpoint of the interval.

26. Example, Page 424 Calculate T N and M N for the value of N indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Example, Page 424 Calculate T N and M N for the value of N indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

28. Example, Page 424 Calculate the approximation to the volume of the solid obtained by rotating the graph about the. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we assume f ″ (x) exists and is continuous on our interval, we may use Theorem 1.

30. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 6 shows how trapezoidal estimates for areas under curves are more accurate for those with small values of f ″.

31. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 6 shows the points we would use in calculating T6 and M6 for an approximation to the area of the shaded region in Figure 8.

32. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 10 illustrates how trapezoids provide an underestimate of areas under concave down curves and midpoints provide over- estimates. The opposite holds true for concave up curves.

33. Example, Page 424 State whether T N or M N overestimates or underestimates the integral and find a bound for the error. Do not calculate for T N or M N. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

34. Example, Page 424 Use the Error Bound to find a value of N for which the Error (T N ) ≤ 10 – 6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

35. Homework Homework Assignment #16 Read Section 7.2 Page 424, Exercises: 1 – 11(Odd), 25, 29, 33, 37 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company