1. Adguary Calwile Laura Rogers Autrey~ 2nd Per. 3/14/11
Finding the area under a curve: Riemann, Trapezoidal, and Simpson’s Rule
Adguary Calwile
Laura Rogers
Autrey~ 2nd Per.
3/14/11

2. Introduction to area under a curve
Before integration was developed, people found the area under curves by dividing the space beneath into rectangles, adding the area, and approximating the answer.
As the number of rectangles, n, increases, so does the accuracy of the area approximation.

3. Introduction to area under a curve (cont.)
There are three methods we can use to find the area under a curve: Riemann sums, the trapezoidal rule, and Simpson’s rule.
For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

4. Finding Area with Riemann Sums
Subintervals with equal width
For convenience, the area of a partition is often divided into subintervals with equal width – in other words, the rectangles all have the same width. (see the diagram to the right for an example of a Right Riemann approximation)

5. Finding Area with Riemann Sums
It is possible to divide a region into different sized rectangles based on an algorithm or rule (see graph above)

6. Riemann Sums
There are three types of Riemann Sums Right Riemann: Left Riemann: Midpoint Riemann:

7. Right Riemann- Overview
Right Riemann places the right point of the rectangles along the curve to find the area. The equation that is used for the RIGHT RIEMANN ALWAYS begins with: And ends with Within the brackets!

8. Right Riemann- Example
Remember: Right Only
Given this problem below, what all do we need to know in order to find the area under the curve using Right Riemann?
4 partitions

9. Right Riemann- Example
For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

10. Right Riemann- Example

11. Right Riemann TRY ME!
Volunteer:___________________
4 Partitions

12. !Show All Your Work!
n=4

13. Did You Get It Right?
n=4

14. Left Riemann- Overview
Left Riemann uses the left corners of rectangles and places them along the curve to find the area. The equation that is used for the LEFT RIEMANN ALWAYS begins with: And ends with Within the brackets!

15. Left Riemann- Example Remember: Left Only
Given this problem below, what all do we need to know in order to find the area under the curve using Left Riemann?
4 partitions

16. Left Riemann- Example For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

17. Left Riemann- Example

18. Volunteer:___________
Left Riemann- TRY ME!
Volunteer:___________
3 Partitions

19. !Show All Your Work!
n=3

20. Did You Get My Answer?
n=3

21. Midpoint Riemann- Overview
Midpoint Riemann uses the midpoint of the rectangles and places them along the curve to find the area. The equation that is used for MIDPOINT RIEMANN ALWAYS begins with: And ends with Within the brackets!

22. Midpoint Riemann- Example
Remember: Midpoint Only
Given this problem below, what all do we need to know in order to find the area under the curve using Midpoint Riemann?
4 partitions

23. Midpoint Riemann- Example
For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

24. Midpoint Riemann- Example

25. Midpoint Riemann- TRY ME
Volunteer:_________
6 partitions

26. !Show Your Work!
n=6

27. Correct???
n=6

28. Applications of Approximating Areas
EXAMPLE
The velocity of a car (in feet per second) is recorded from the speedometer every 10 seconds, beginning 5 seconds after the car starts to move. See Table 2. Use a Riemann sum to estimate the distance the car travels during the first 60 seconds. (Note: Each velocity is given at the middle of a 10-second interval. The first interval extends from 0 to 10, and so on.)
SOLUTION
Since measurements of the car’s velocity were taken every ten seconds, we will use Now, upon seeing the graph of the car’s velocity, we can construct a Riemann sum to estimate how far the car traveled.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #26

29. Applications of Approximating Areas
This is an example of using a midpoint Riemann sum to approximate an integral.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #27

30. Applications of Approximating Areas
CONTINUED
Therefore, we estimate that the distance the car traveled is 2800 feet.
Goldstein/SCHNIEDER/LAY, CALCULUS AND ITS APPLICATIONS, 11e – Slide #28

31. Trapezoidal Rule Overview
Trapezoidal Rule is a little more accurate than Riemann Sums because it uses trapezoids instead of rectangles. You have to know the same 3 things as Riemann but the equation that is used for TRAPEZOIDAL RULE ALWAYS begins with: and ends with Within the brackets with every“ f ” being multiplied by 2 EXCEPT for the first and last terms

32. Trapezoidal Rule- Example
Remember: Trapezoidal Rule Only
Given this problem below, what all do we need to know in order to find the area under the curve using Trapezoidal Rule?
4 partitions

33. Trapezoidal Example For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

34. Trapezoidal Rule- Example

35. Trapezoidal Rule- TRY Me
Volunteer:_____________
4 Partitions

36. Trapezoidal Rule- TRY ME!!
n=4

37. Was this your answer?
n=4

38. Simpson’s Rule- Overview
Simpson’s rule is the most accurate method of finding the area under a curve. It is better than the trapezoidal rule because instead of using straight lines to model the curve, it uses parabolic arches to approximate each part of the curve. The equation that is used for Simpson’s Rule ALWAYS begins with:
And ends with
Within the brackets with every “f” being multiplied by alternating coefficients of 4 and 2 EXCEPT the first and last terms.
In Simpson’s Rule, n MUST be even.

39. Simpson’s Rule- Example
Remember: Simpson’s Rule Only
Given this problem below, what all do we need to know in order to find the area under the curve using Simpson’s Rule?
4 Partitions

40. Simpson’s Example For each method we must know:
f(x)- the function of the curve
n- the number of partitions or rectangles
(a, b)- the boundaries on the x-axis between which we are finding the area

41. Simpson’s Rule- Example

42. Simpson’s Rule TRY ME!
Volunteer:____________
4 partitions

43. !Show Your Work!
n=4

44. Check Your Answer!

45. Sources http://www.intmath.com/Integration
© Laura Rogers, Adguary Calwile; 2011