1. Numerical Integration
Lesson 6.5

2. Trapezoidal Rule
Instead of calculating approximation rectangles we will use trapezoids
More accuracy
Area of a trapezoid a b •
Which dimension is the h?
Which is the b1 and the b2 b1 b2 h

3. Trapezoidal Rule Trapezoidal rule approximates the integraldx
f(xi)
f(xi-1)
Trapezoidal rule approximates the integral
Calculator function for f(x) ((2*f(a+k*(b-a)/n),k,1,n-1)+f(a)+f(b))*(b-a)/(n*2) trap(a,b,n)

4. Trapezoidal Rule Entering the trapezoidal rule into the calculator
f(x) must be defined for this to work

5. Trapezoidal Rule
Try using the trapezoidal rule
Check with integration

6. Snidly Fizbane Simpson
Simpson's Rule
As before, we divide the interval into n parts
n must be even
Instead of straight lines we draw parabolas through each group of three consecutive points
This approximates the original curve for finding definite integral – formula shown below
Snidly Fizbane Simpson a b •

7. Simpson's Rule Our calculator can do this for us also
The function is more than a one liner
We will use the program editor
Choose APPS, 7:Program Editor 3:New
Specify Function, name it simp

8. Simpson's Rule
Enter the parameters a, b, and n between the parentheses
Local variables discarded when function finishes
Initialize dx
Enter commands shown between Func and endFunc
Initialize total with the two end values
One for loop for the 4* values, one for the 2* values
Return the value

9. Simpson's Rule Specify a function for f(x) When you call simp(a,b,n),
Make sure n is an even number
Note the accuracy of the approximation

10. Using Data
Given table of data, use trapezoidal rule to determine area under the curve
dx = ? x
2.00
2.10
2.20
2.30
2.40
2.50
2.60 y
4.32
4.57
5.14
5.78
6.84
6.62
6.51

11. Using Data
Given table of data, use Simpson's rule to determine area under the curve x
2.00
2.10
2.20
2.30
2.40
2.50
2.60 y
4.32
4.57
5.14
5.78
6.84
6.62
6.51

12. Assignment
Lesson 6.5
Page 250
Exercises 1 – 21 odd