1. AP Calculus BC Friday, 30 January 2015 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS are not graded. Get an 89 calculator and an 84 calculator. AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

2. Sec. 7.1: Area of a Region Between Two Curves Curves may give different regions. Ex:Find the bounded area between f (x) = –3x 3 – 2x 2 + 5x and g (x) = –3x (–2, 6) (0, 0) ( 4 / 3, –4) f (x) g (x)

3. Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1 (–1, –2) (4, 3) (–5, 0) x = y 2 – 5 y = x – 1

4. Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1. (–1, –2) (4, 3) (–5, 0)

5. Stopped here 1 st Period on Monday, 03 February 2014. Stopped here 2 nd Period on Monday, 03 February 2014.

6. Sec. 7.1: Area of a Region Between Two Curves Ex:Find the area of the region bounded by x = y 2 – 5 and y = x – 1 It is sometime advantageous to integrate with respect to y. (–1, –2) (4, 3) (–5, 0) x = y 2 – 5 y = x – 1

7. Sec. 7.1: Area of a Region Between Two Curves In general – Vertical Rectangles Horizontal Rectangles

8. Sec. 7.1: Area of a Region Between Two Curves The consumption (in billions of barrels) of petroleum in the United States from 1960 to 1979 can be modeled by where t = 0 corresponds to January 1, 1970. From 1979 through 1990, the consumption can be modeled by Find the total amount of fuel saved from 1979 through 1990 as a result of fuel being consumed at the post 1979 rate rather than at the pre-1979 rate.

9. Sec. 7.1: Area of a Region Between Two Curves t = 9t = 20 The total amount of fuel saved from 1979 through 1990 was 21.674 billion barrels of petroleum.

10. AP Calculus BC Tuesday, 31 January 2012 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS are not graded. YOU WILL NEED A TI-89 (from the table) AND A TI-84. –Sixth Period will need to install batteries.

11. Sec. 7.2: Volume: The Disk Method

12. Some examples of solids of revolution:

13. Sec. 7.2: Volume: The Disk Method If a region in the plane is revolved about a line, the resulting solid is a solid of revolution.

14. Sec. 7.2: Volume: The Disk Method When we put these disks together, we get a solid.

15. Stopped here 2 nd Period on Friday, 30 January 2015.

16. AP Calculus BC Monday, 02 February 2015 OBJECTIVE TSW (1) find the volume of a solid of revolution using the disk method, and (2) find the volume of a solid of revolution using the washer method. TESTS are graded. ASSIGNMENT DUE DATES –Sec. 7.1  Tuesday, 02/03/15 –Sec. 7.2: Disc Method  Wednesday/Thursday, 02/04- 05/15 –Sec. 7.2: Washer Method  Wednesday/Thurday, 02/04- 05/15 AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

17. Sec. 7.2: Volume: The Disk Method

18. Ex:Find the volume of the solid of revolution by rotating about the x-axis to the line x = 9. x = 9 (0, 0)

19. AP Calculus BC Monday, 02 February 2015 OBJECTIVE TSW (1) find the volume of a solid of revolution using the disk method, and (2) find the volume of a solid of revolution using the washer method. TESTS are graded. ASSIGNMENT DUE DATES –Sec. 7.1  Tuesday, 02/03/15 –Sec. 7.2: Disc Method  Wednesday/Thursday, 02/04- 05/15 –Sec. 7.2: Washer Method  Wednesday/Thurday, 02/04- 05/15 AP EXAM REGISTRATION (through 03/04/15) www.TotalRegistration.net/AP/443381

20. Stopped here 1 st Period on Friday, 30 January 2015.

21. Sec. 7.2: Volume: The Disk Method Ex:Find the volume of the solid of revolution by rotating the region formed by y = 3 – x 2 and y = 2 about the line y = 2. y = 2 y = 3 – x 2 (–1, 2) (1, 2)

22. Ex:Find the volume of the solid of revolution by rotating the bounded region formed by f (x) = x 3, the y-axis, and y = 8 about the y-axis. Sec. 7.2: Volume: The Disk Method y = 8 f (x) = x 3 (0, 0) (2, 8)

23. Jersey Village Falcons

24. AP Calculus BC Friday, 08 February 2013 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. TESTS will be given back Monday.

25. Stopped here 2 nd Period on Thursday, 07 February 2013.

26. AP Calculus BC Thursday, 06 February 2014 OBJECTIVE TSW (1) find the volume of a solid of revolution using the washer method, and (2) find the volume of a solid with known cross sections. TESTS are not graded. You will need a TI-83/84 (mine or yours). AP EXAM REGISTRATION (through 03/07/14) www.TotalRegistration.net/AP/443381

27. Sec. 7.2: Volume: The Washer Method

28. Stopped here 2 nd Period on Monday, 02 February 2015.

29. NOTE: To link to SketchPad on the next slide, the program AND file must be already open.

30. Sec. 7.2: Volume: The Washer Method Sketchpad: Solids of Revolution

31. Sec. 7.2: Volume: The Washer Method

32. AP Calculus BC Monday, 06 February 2012 OBJECTIVE TSW (1) find the area of a region between two curves; (2) find the area of a region between intersecting curves; and (3) describe integration as an accumulation process. You will need a TI-89 and a TI-84.

33. Sec. 7.2: Volume: The Washer Method Ex:Find the volume of the solid formed by rotating the bounded region of and y = x 2 about the x-axis.

34. Stopped after the previous problem for 6 th period on Friday, 03 February 2012.

35. Stopped after the previous problem for 7 th period on Wednesday, 01 February 2012.

36. Sec. 7.2: Volume: The Washer Method The Washer Method If R(x) is the outer radius and r(x) is the inner radius of a solid of revolution, then the volume of the solid of revolution on [a, b] is Another way to look at this is as the Marshall Plan: named in honor of Ashley Marshall, JVHS c/o 1999.

37. Showed the slide (just the start) of the next problem 1 st Period on Wednesday, 06 February 2013.

38. Sec. 7.2: Volume: The Washer Method Ex:Find the volume of the solid formed by rotating the bounded region of and y = x/4 about the x-axis.

39. Stopped here 1 st Period on Monday, 02 February 2015.

40. Sec. 7.2: Volume: The Washer Method On a blank sheet of paper (to be turned in), for each problem, (a) draw a picture (label, shade, representative rectangle, axis of rotation), (b) set up the integral, and (c) evaluate. Use exact values, leaving π in your answer.  wire basket Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. (1)the x-axis (2)the y-axis (3)the line x = 4 (4)the line x = 6

41. Sec. 7.2: Volume: The Washer Method On a blank sheet of paper (to be turned in), for each problem, (a) draw a picture (label, shade, representative rectangle, axis of rotation), (b) set up the integral, and (c) evaluate. Use exact values, leaving π in your answer.  wire basket Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. (1)the x-axis(2)the y-axis (3)the line x = 4(4)the line x = 6