Riemann Sums, Trapezoidal Rule, & Simpson’s Rule

1. Riemann Sums, Trapezoidal Rule, & Simpson’s Rule By: Carson Smith & Elisha Farley

2. Riemann Sums • A Riemann sum is a method for approximating the total area underneath a curve on a graph. • This method is also known as taking an integral. • There are 3 forms of Riemann Sums: Left, Right, and Middle.

3. Right Riemann Riemann Sums Illustrated Middle Riemann Left Riemann

4. Riemann Sum Formula B To find the intervals needed, use the formula: A Where B = the upper limit, A = the lower limit, and N = the number of rectangles used. N = 4

5. Riemann Sum Formula Cont. Then incorporate the previous intervals into the formula:

6. Left Riemann Example For a Left Riemann, use all of the functions except for the last one. The Left Riemann under approximates the area under the curve.

7. Right Riemann Example For a Right Riemann, use all of the functions except for the last one. The Right Riemann over approximates the area under the curve.

8. Middle Riemann Example For a Middle Riemann, average all the intervals found and plug the averages into the functions. The Middle Riemann is the closest approximation.

9. Integration Answer The Middle Riemann is the closest approximation

10. Try A Left Riemann! N = 4

11. Left Riemann Solution N = 4

12. Riemann Sum Program Usage • Click the “PRGM” button. • Select the RIEMANN program. • Enter your f(x). • Enter Lower & Upper bounds. • Enter Partitions • Select Left, Right, or Midpoint Sum

13. Trapezoidal Rule • Like Riemann Sums, Trapezoidal Rule approximates the are under the curve using trapezoids instead of rectangles to better approximate.

14. Trapezoidal Rule Illustrated

15. Trapezoidal Rule Formula • Use the same formula to find your intervals. • Then plug your intervals into the equation:

16. Trapezoidal Rule Example N = 4

17. Trapezoidal Rule Example Cont. Remember to multiply all intervals by 2, excluding the first and last interval.

18. Try This Trapezoidal Rule Problem! N = 4

19. Trapezoidal Rule Solution N = 4

20. Trapezoidal Rule Program Usage • Click the “PRGM” button. • Select the RIEMANN program. • Enter your f(x). • Enter Lower & Upper bounds. • Enter Partitions • Select Trapezoid Sum

21. Simpson’s Rule • Simpson’s rule, created by Thomas Simpson, is the most accurate approximation of the area under a curve as it uses quadratic polynomials instead of rectangles or trapezoids.

22. Simpson’s Rule Formula Simpson’s Rule can ONLY be used when there are an even number of partitions. Still use the formula: to find your intervals to plug into the equation.

23. Simpson’s Rule Example N = 4

24. Simpson’s Rule Example Cont. When using Simpson’s Rule, multiply all intervals excluding the first and the last alternately between 4 & 2, always starting with 4

25. Try This Simpson’s Rule Problem!

26. Simpson’s Rule Solution

27. Simpson’s Rule Program Usage • Click the “PRGM” button. • Select the SIMPSON program. • Enter Lower & Upper bounds. • Enter your N/2 Partitions. • Enter your f(x)

28. 1994 AB 6

29. 1994 AB 6 “A” Solution

30. 1994 AB 6 “B” Solution

31. 1994 AB 6 “C” Solution