Numerical Integration Lesson 5.6

News from the Internet 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!

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

Trapezoidal Rule Trapezoidal rule approximates the integral dx 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)

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

Trapezoidal Rule Try using the trapezoidal rule Check with integration

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 •

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

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

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

Assignment A Lesson 5.6 Page 350 Exercises 1 – 23 odd

Error Estimation Trapezoidal error for f on [a, b] Where M = max value of |f ''(x)| on [a, b] Simpson's error for f on [a, b] Where K = max value of |f (4)(x)| on [a, b]

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

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

Assignment B Lesson 5.6 Page 350 Exercises 27 – 39 odd 49, 51, 53