2. Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2 only e) -1, 0, and 2 only

3. Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2 only e) -1, 0, and 2 only

4. Chap. 4: Integral Calculus Introduction ©2002 Roy L. Gover (www.mrgover.com) Connection between differential and integral calculus Example of an integral calculus application

5. Important Idea Differential Calculus solves the tangent line problem Instantaneous rate of change Related rates Maxima & minima Optimization

6. Important Idea Integral Calculus solves the area problem Area of a region Volumes of solids Work, force & fluid pressure Accumulated rates of change

7. The tangent line problem and the area problem are related… one is the inverse of the other Important Idea

8. “This discovery (called the Fundamental Theorem of Calculus) brought differential and integral calculus together to become the single most powerful insight mathematicians had for understanding how the universe works.” -Dan Kennedy, textbook author

9. Example Velocity is the tangent line at any point on the position curve. But...

10. Area is Distance d=vt Given a constant velocity of 40 mph, find distance.

11. Distance is area even when velocity is not constant.

12. Distance is area even if the velocity function is complex

13. For the remainder of this course, we will focus on ways to solve problems involving area under a curve. The process of finding area is called antidifferentiation or integration.

14. 4-2: Sigma Notation & Area Objectives: Use sigma notation to write and evaluate a sum. Use sigma notation in finding area. ©2002 Roy L. Gover (www.mrgover.com)

15. Connection In the last lesson you learned how to find an antiderivative. In this lesson you will use sigma notation to find area.

16. Connection In a future lesson, you will learn that these ideas are closely related by an important theorem called the Fundamental Theorem of Calculus.

17. Example Find the sum:

18. Definition Index Sigma Notation Upper Bound i th Term Lower Bound

19. Definition This summation is called a Riemann sum. It is used to estimate the area under a curve. More on this later…

20. Important Idea Rectangles can be placed under a graph. The sum of the areas of the rectangles approximate the area under the graph. Right End Point Approx Upper Sum

21. Important Idea Rectangles can be placed under a graph. The sum of the areas of the rectangles approximate the area under the graph. Left End Point Approx Lower Sum

22. Important Idea Rectangles can be placed under a graph. The sum of the areas of the rectangles approximate the area under the graph. Mid Point Approx

23. Which method gives the best approximation and why?

24. What could be done to get a better approximation? Go To AREAPPR

25. Definition i th Rectangle mimi f(m i ) A representative rectangle:What is its area?

26. Example Use 4 rectangles to find a right end point approx. of the area under the graph between x =0 & x =2 0 2

27. Try This Use 4 rectangles to find a left end point approx. of the area under the graph between x =0 & x =2 0 2 6.25 sq. units

28. Try This Use 4 rectangles to find a mid point approximation of the area under the graph between x =0 & x =2 0 2 7.875 sq. un.

29. Important Idea As the number of rectangles used to approximate an area gets large, the width of each rectangle gets small and the error of the area approximation approaches zero.

30. Definition A Riemann Sum to find area using a mid point approx.: where x = m i is the mid point of the i th rectangle

31. Definition A Riemann Sum to find area using a left end pt. approx.: where x = l i is the left end point of the i th rectangle

32. Definition A Riemann Sum to find area using a right end pt. approx.: where x = r i is the right end point of the i th rectangle

33. Example Use geometry to find the area between f(x) = x and the x axis between x =0 & x =2. 2

34. Lesson Close From memory, list the formulas for find the exact area using a Riemann Sum.

35. Assignment Area Worksheet Problems 1- 4 (#1 only on 4)