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

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Presentation on theme: "Finding the area under a curve: Riemann, Trapezoidal, and Simpsons Rule Adguary Calwile Laura Rogers Autrey~ 2 nd Per. 3/14/11."— Presentation transcript:

1 Finding the area under a curve: Riemann, Trapezoidal, and Simpsons Rule Adguary Calwile Laura Rogers Autrey~ 2 nd 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 Simpsons 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 Riemann Sums There are three types of Riemann Sums Right Riemann: Left Riemann: Midpoint Riemann:

5 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!

6 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

7 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

8 Right Riemann- Example

9 Right Riemann TRY ME! Volunteer :___________________ 4 Partitions

10 !Show All Your Work! n=4

11 Did You Get It Right? n=4

12 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!

13 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

14 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

15 Left Riemann- Example

16 Left Riemann- TRY ME! Volunteer:___________ 3 Partitions

17 !Show All Your Work! n=3

18 Did You Get My Answer? n=3

19 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!

20 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

21 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

22 Midpoint Riemann- Example

23 Midpoint Riemann- TRY ME 6 partitions Volunteer:_________

24 !Show Your Work! n=6

25 Correct??? n=6

26 Trapezoidal Rule Overview Trapezoidal Rule is a little more accurate that 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

27 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

28 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

29 Trapezoidal Rule- Example

30 Trapezoidal Rule- TRY Me Volunteer:_____________ 4 Partitions

31 Trapezoidal Rule- TRY ME!! n=4

32 Was this your answer? n=4

33 Simpsons Rule- Overview Simpsons 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 Simpsons 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 Simpsons Rule, n MUST be even.

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

35 Simpsons 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

36 Simpsons Rule- Example

37 Simpsons Rule TRY ME! 4 partitions Volunteer:____________

38 !Show Your Work! n=4

39 Check Your Answer!

40 Sources © Laura Rogers, Adguary Calwile; 2011


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