1. NUMERICAL INTEGRATION

2. Mid-point RULE
We will divide the interval [a, b] into n subintervals of equal width,
We will denote each of the intervals as follows,
where and

3. Mid-point RULE
Then for each interval let  be the midpoint of the interval.  We then sketch in rectangles for each subinterval with a height of Here is a graph showing the set up using n=6

4. Mid-point RULE

5. Mid-point RULE
We can easily find the area for each of these rectangles and so for a general n we get that,
Or, upon factoring out  we get the general Rectangular Rule:

6. TRAPEZOIDAL RULE
For this rule we will do the same set up as for the Rectangular Rule.  We will break up the interval [a, b]into n subintervals of width,
Then on each subinterval we will approximate the function with a straight line that is equal to the function values at either endpoint of the interval.  Here is a sketch of this case for n=6 .

7. TRAPEZOIDAL RULE

8. TRAPEZOIDAL RULE The area of the trapezoid in the interval is given by
So, if we use n subintervals the integral is approximately,

9. TRAPEZOIDAL RULE
Simplifying the equation, gives us the Trapezoidal rule as

10. SIMPSON’S RULE
We will again divide up the interval [a, b] into n subintervals. The width of each subinterval is,
In the Trapezoidal Rule we approximated the curve with a straight line.  For Simpson’s Rule we are going to approximate the function with a quadratic equation

11. SIMPSON’S RULE

12. SIMPSON’S RULE
Notice that each approximation actually covers two of the subintervals.  So for Simpson’s rule we require n to be even.  It can be shown that the area under the approximation on the intervals  and is,

13. SIMPSON’S RULE
If we use n subintervals the integral is then approximately,
The general form of Simpson’s rule is

14. Examples i
xi*
f(xi*) 1
1.125
1.42 2
1.375
2.60 3
1.625
4.29 4
1.875
6.59

15. Trapezoidal Rule Simpson’s Rule i xi f(xi) 1 1.25 1.95 2 1.5 3.38 3
1.75
5.36 8
Simpson’s Rule

16. Extra practice sums Apply Simpson’s rule to evaluate with n = 6.
Evaluate using Simpson’s rule with n = 4.
Evaluate using Trapezoidal rule with
n = 5.