Numerical Integration

1. Numerical Integration Lesson 6.5

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

3. a b • Which dimension is the h? • Which is the b1 and the b2 b2 h b1 Trapezoidal Rule • Instead of calculatingapproximation rectangleswe will use trapezoids • More accuracy • Area of a trapezoid •

4. f(xi-1) f(xi) dx 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)

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

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 Editor3:New • Specify Function,name it simp

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

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 = ?

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

13. Assignment • Lesson 6.5 • Page 250 • Exercises 1 – 21 odd