1. Integration

2. First some review… To integrate, you must have a sum/difference or constant multiple combination of a variable to a number power (x n ). You may need to substitute one function for another in order to manipulate the integral to be in this form! Integration finds net area between curves by accumulating heights.

3. Now you’re ready to play! Get into the following groups of students – move your desks into these groups!! E. Yoder D. Parker K. Campbell M. Flannery J. Andrejko R. Bauters J. Jurado D. Silva G. Hucks

4. Now you’re ready to play! Designate one person in your group to “keep score”. This person should draw a blank Tic-Tac-Toe board on a piece of paper. Keep track of which spaces you have filled in on this board by marking an X for the ones you get correct and an O for the ones do you not. Remember: Your goal is to make three X’s across, down, or diagonally (just like in normal Tic-Tac-Toe). First group to accomplish this, wins 10 bonus points!

5. Choose one square of each type to make Tic-Tac-Toe… Comprehensio n Creative Thinking Application Creative Thinking Application Comprehensio n Application Comprehensio n Creative Thinking

6. Comprehension Integrate: (Do not click the option below until EVERYONE has attempted the problem!) Answer

7. Comprehension Answer Discuss as a class why the answer is negative. Hint: Think about the location of the graph of cosx between π / 2 and π. Back to Game Board

8. Comprehension Integrate: (Do not click the option below until EVERYONE has attempted the problem!) Answer

9. Comprehension Answer Back to Game Board

10. Comprehension Integrate (Do not click an option below until EVERYONE has attempted the problem!) Answer

11. Comprehension Answer Back to Game Board

12. Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

13. Creative Thinking Answer Back to Game Board

14. Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

15. Creative Thinking Answer Back to Game Board

16. Creative Thinking Integrate (Do not click the option below until EVERYONE has attempted the problem!) Answer

17. Creative Thinking Answer Back to Game Board Recall…

18. Application The graph of f(x) is shown: Calculate (Do not click the option below until EVERYONE has attempted the problem!) Answer

19. Application Answer Recall…the area above the x-axis is positive and the area below the x-axis is negative in integration. Using geometric shapes, divide the positive region into a triangle from [-1,0], a rectangle from [0,1], and a triangle from [1,2]. Add these up! 1+2+1 = 4 Subtract the triangular region from [2,3] and the rectangular region from [3,4]. 4-.5-1 = 2.5 Back to Game Board

20. Application The function f is continuous for all 1 < x < 7. Approximate the value of using MRAM with 3 equal subdivisions. (Do not click the option below until EVERYONE has attempted the problem!) Answer x1234567 f(x)123.5200

21. Application Answer Since we need 3 equal subdivisions, use: A 1 = the area from [1,3], ∆x = 2 A 2 = the area from [3,5], ∆x = 2 and A 3 = the area from [5,7], ∆x = 2. A 1 = (2)(f(2)) = 2(2) = 4 A 2 = (2)( f(4))= 2(2) = 4 A 3 = (2)( f(6)) = 2(-1) = -2 A 1 + A 2 + A 3 = 4+4+-2 = 6. Back to Game Board x1234567 f(x)123.5200

22. Application The temperature T (in ˚C) recorded during a day followed the curve where t is time and noon is t = 0. (In other words, for this day, -12 ≤ t ≤ 12.) What was the average temperature during the day? (Do not click the option below until EVERYONE has attempted the problem!) Answer

23. Application Answer For average temperature, “add up all the temperatures and divide by the number of temperatures accumulated.” Back to Game Board