1. 6.1 Area Between 2 Curves Wed March 18 Do Now Find the area under each curve in the interval [0,1] 1) 2)

2. Graphing Calculator - Integrals This only works if you have bounds Math -> 9 fnint( Fnint(equation,x,lower,upper)

3. Test Review Retakes by?

4. Area Between 2 Curves If f(x) > g(x) on the interval [a,b], then the area between f(x) and g(x) on the interval [a,b] is

5. Area Between 2 Curves 1) Graph both functions 2) Decide which function is f(x) and which is g(x) 3) Evaluate each integral on the given interval 4) Subtract the 2 values

6. Examples Find the area between the curves on the interval [0,2]

7. Example 2 Find the area between the curves on the interval [1,3]

8. Area between 2 Curves that Cross If the curves cross inside a given interval, we need to split it up at the intersection point. Set the 2 functions equal to each other to find the intersection point. –Can use the graphing calculator to find intersection points

9. Ex Ex: Find the area bounded by the graphs on the interval [0,2]

10. You try 1) Find the area of the region in the interval [1, 3] between the functions 2) Find the area in the interval [-2, 5] between the functions

11. Closure Hand in: Find the area between the curves on the interval [0,2] HW: p.361-2 #1,2, 5, 8, 9, 16

12. 6.1 Area Bounded by Curves Thurs March 19 Do Now Find the area between the 2 curves on the [0,5] 1) 2)

13. HW Review: p.361 #1,2, 5, 8, 9, 16 1) 102 2) 34/3 = 11.333 5) 8) 262/3 = 87.333 9) 17) 2 – pi/2 =.429

14. Area Between 2 Curves that Intersect (no given interval) 1) Set f(x) = g(x) 2) Solve for x –The two x values are your lower and upper bounds 3) Evaluate the area between the 2 curves Calculator: use graphs to find intersection points

15. Using a Calculator Graph both functions 2 nd -> calc -> intersect Pick the 2 curves you want to find the intersection of Guess: pick a point near the intersection point

16. Examples Find the area bounded by the graphs of

17. Ex 2 Find the area bounded by the graphs of

18. Areas determined by 3 Curves If the upper or lower curve changes from one function to another, we split it up into 2 or more areas Ex:

19. You try Find the area of the region that is enclosed between the curves

20. You try Find the area bounded by the graphs of

21. Closure Hand in: Find the area of the region that is enclosed between the curves HW: p.361 #3 4 10 13 27 29 31

22. 6.1 Area Practice Fri March 20 Do Now Calculate the area determined by the intersections of the curves

23. HW Review: p. 361 #3 4 10 13 27 29 31 3) 32/3 = 10.667 4) 128 10) 12ln6 – 10 = 11.501 13) 160/3 = 53.333 27) 64/3 = 21.333 29) 2 31) 128/3 = 42.667

24. Practice (green book) worksheet p.409 #7-12, 15-20, 30-34

25. Closure Find the area enclosed by the following HW: worksheet p.409 #7-12 15-20 30- 34

26. 6.1 Areas with Respect to Y Mon March 23 Do Now Find the area bounded by the following functions

27. HW Review: p.409 #7-12 7)4.86 8)20.65 9) 3 10) 31/6 = 5.167 11) 29/2 = 14.5 12) 12

28. HW Review: p.409 #15-20 15) 1/6 =.167 16) 36 17) 27/4 = 6.75 18) 27/4 = 6.75 19) 1/12 =.083 20) 1/3 =.333

29. HW Review: p.409 #30-34 30) 4/3 31) 1 32) 4 33) 8 34) 32/3

30. Integrating with Respect to Y If f(y) is to the right of g(y), then the area between two curves is 1) Set all functions x = f(y) and x = g(y) 2) The curve on the right is f(y) 3) Evaluate the areas and subtract

31. Revisiting Previous Example Let’s integrate with respect to y instead of splitting the area up into 2 areas.

32. Example 1.6 Find the area bounded by the graphs of

33. Closure Hand in: Sketch and find the area bounded by the given curves. Choose the variable of integration so that the area is written as a single integral HW: p.363 #19-26 skip 24 Quiz Fri

34. 6.1 Area Review Tues March 24 Do Now Sketch and find the area of the region bounded by the given curves.

35. HW Review: p.363 #19-26 19) 1331/6 = 221.833 20) 64/3 = 21.333 21) 256 22) 81/2 = 40.5 23) 32/3 25) 64/4 26) 3

36. Area Review Area between 2 curves F(x) - G(x) Finding intersection points = bounds Area inside 3 curves –Splitting up into 2 areas What are the bounds of each area? –Integrating with respect to Y Using Y - bounds, and Y - functions

37. Practice (blue book) Worksheet p.448 #1-4, 7-13 odds

38. Closure Journal Entry: When trying to find the area between two curves, when should we integrate with respect to x? to y? Would you rather switch between x and y, or have to split your area problem into several problems? Why? HW: Finish worksheet p.448 #1-6 all, 7-13 odds Quiz Fri March 27

39. 6.1 Quiz Review Wed March 25 Do Now Find the area of the region R between the curves 1) y = 2x + 1, y = 0, y = -x 2) y = x^2, y = -x + 2, y = 0

40. HW Review: worksheet p.448 #1-6, 7-13 odds 1) 9/211) sqrt 2 2) 22/313) 1/2 3) 119) 37/12 4) 10/321) ~5.66 5) 32/3 6) 9 7) 49/192 9) 1/2

41. Area Review Area between 2 curves F(x) - G(x) Finding intersection points = bounds Area inside 3 curves –Splitting up into 2 areas What are the bounds of each area? –Integrating with respect to Y Using Y - bounds, and Y - functions

42. Quiz All types of areas Can use graphing calculator, but must set up the integral correctly first Show all work used to set up the integral

43. Practice (oj) worksheet p.395 #1-6 9 10 13-17

44. Closure How can you find the area between 2 or more curves? How can you find the bounds? HW: Finish worksheet p.395 #1-10 13- 17 6.1 Quiz Fri

45. Area Review Thurs March 26 Do Now Find the area bounded by the following: Y = 8x – 10 Y = x^2 - 4x +10

46. Area 1) Graph all the functions 2) Shade the region bounded by the functions 3) If possible, split up into several areas 4) For each area: Integrate higher – lower –Bounds = intersection points (smallest – largest) –Calculator

47. HW Review: worksheet p.395 #1-10 13-17 1) 1.57110) 0.833 2) 4.18913) 16 3).083314) 8.167 4) 1.33315) 10.667 5) 8.53316) 10.667 6) 1.46717) 4 9) 0.833

48. Area Review 6 problems Area between 2 curves Higher curve – lower curve Finding intersection points = bounds Area inside 3 curves –Splitting up into 2 areas What are the bounds of each area? –Integrating with respect to Y Using Y - bounds, and Y - functions

49. You try The area bounded by y = sqrt x, y = -sqrt x, and y = 1 - x

50. Quiz All types of areas Can use graphing calculator, but must set up the integral correctly first Show all work used to set up the integral

51. Closure How can you find the area between 2 or more curves? How can you find the bounds? When would you integrate with respect to x or y? 6.1 Quiz tomorrow