1. Numerical Integration
AP Calculus

2. Numerical Integration
* Used when normal definite integration is not possible.
a). When there is no elementary function for the anti-derivative;
i.e.: or
b). Data is given in tabular or graphical form and it is too much
effort to find the representative function.

3. I. RIEMANN’S SUMS
REM: Riemann’s Sum uses Rectangles to approximate the
accumulation.
A = bh => h - Left Endpoints Right Endpoints
- Midpoint
The more accurate is the Midpoint Sum
(must remember how to use all three – Left, Right, and Midpoint)
Midpoint Rule: with
In Words: The width of the subinterval times the sum of the heights
AT THE MIDPOINT of each subinterval.

4. Illustration:
Function:
  on [ 0 , 4 ] with n = 4

5. Example: Graphical
Find the Average Revenue for the 5 years.

6. II. TRAPEZOID METHOD: rather than rectangles: REM:
 Uses Trapezoids to fill the regions
rather than rectangles:
REM:
(Notice this is the average of the Left and Right Riemann's Sums)

7. Trapezoid Rule:
 In Words: One half * width of subinterval * the ( 1 , 2 , 2 , … , 2 , 1 ) pattern of the heights found at the points of the subinterval.

8. Illustration: (Trapezoid)
Function:
  on [ 0 , 4 ] with n = 4

10. t (sec) 3 6 9 12 15 a(t) (ft/sec2) 4 8 10 Example: Data
The data for the acceleration a(t) of a car from 1 to 15 seconds are given in the table below. If the velocity at t = 0 is 5 ft/sec, which of the following gives the approximate velocity at t = 15 using the Trapezoidal Rule?
t (sec) 3 6 9 12 15
a(t) (ft/sec2) 4 8 10

11. A lot is bounded by a stream
A lot is bounded by a stream. and two straight roads that meet at right angles. Use the Trapezoid Rule to approximate the area of the lot (x and y are measured in square meters)

12. Built on: The area of the region below a quadratic function.
III. SIMPSON’S METHOD
Built on: The area of the region below a
quadratic function.
REM: Three points are required to write a quadratic equation
since the equation has 3 variables; A,B,C in
Therefore, to get the 3 points needed, Simpson’s uses double subintervals to
approximate the accumulation.
THEOREM:
with n even

13. __________________________________________________
Simpson’s: (cont)
EXTENDED:
__________________________________________________
Simpson’s Formula:
Note the pattern: 1,4,2,4,1
1,4,2,4,2,4,1 etc

14. Illustration: (Simpson’s)
Function:
  on [ 0 , 4 ] with n = 4

15. Although the economy is continuously changing, we analyze it with discrete measurements. The following table records the annual inflation rate as measured each month for 13 consecutive months. Use Simpson’s Rule with
n = 12 to find the overall inflation rate for the year.
Month
Annual Rate
January
0.04
February
March
0.05
April
0.06
May
June
July
August
September
October
November
December

16. Example: Graphical - all three

17. Error:
Approximation gives rise to two questions>>>>
1) How close are we to the actual answer? and
2) How do we get close enough?

18. Error: MIDPOINT
Error using Midpoint with n partitions
C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”
Where Mi  the MAX of
on i[a,b]

19. Error: TRAPEZIOD
Error using Trapezoid with n partitions
C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”
Where Mi  the MAX of
on i[a,b]

20. Example: How close are we?
Approximate using Trapezoid Method
with 4 intervals and find the Error bound.

21. Example: How many intervals are required?
Approximate using Midpoint Rule
to within .

22. Error: SIMPSONS
Error using Simpson’s with n partitions
C is an un-findable number in [a,b] but whose existence is guaranteed; therefore do “ERROR BOUND”
Where Mi  the MAX of
on i[a,b]

23. Last Update:
02/05/10