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Numerical Integration

Numerical Integration. Lesson 6.5. a. b. Which dimension is the h? Which is the b 1 and the b 2. b 2. h. b 1. Trapezoidal Rule. Instead of calculating approximation rectangles we will use trapezoids More accuracy Area of a trapezoid. •. f(x i-1 ). f(x i ). dx. Trapezoidal Rule.

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Numerical Integration

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  1. Numerical Integration Lesson 6.5

  2. 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 •

  3. 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)

  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. 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 •

  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 Editor3:New • Specify Function,name it simp

  8. 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

  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 = ?

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

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

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