Numerical Integration Lesson 6.5

2 News from Space A new species has been trapped … the rare zoid Math students have long known of efforts of "trapezoid" expeditions Trapezoids were so effective, zoids were thought to be extinct!

3 Trapezoidal Rule Instead of calculating approximation rectangles we will use trapezoids  More accuracy Area of a trapezoid a b b1b1 b2b2 h Which dimension is the h? Which is the b 1 and the b 2 Which dimension is the h? Which is the b 1 and the b 2

4 Trapezoidal Rule 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) dx f(x i ) f(x i-1 )

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

6 Trapezoidal Rule Try using the trapezoidal rule Check with integration

7 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

8 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

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

10 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

11 Using Data Given table of data, use trapezoidal rule to determine area under the curve  dx = ? x y

12 Using Data Given table of data, use Simpson's rule to determine area under the curve x y

13 Assignment Lesson 6.5 Page 250 Exercises 1 – 21 odd