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Numerical Integration. Lesson 6.5. 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!. a. b. Which dimension is the h?

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news from space
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!
trapezoidal rule

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

trapezoidal rule1

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)
trapezoidal rule2
Trapezoidal Rule
  • Entering the trapezoidal rule into the calculator
  • f(x) must be defined for this to work
trapezoidal rule3
Trapezoidal Rule
  • Try using the trapezoidal rule
  • Check with integration
simpson s rule

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

simpson s rule1
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
simpson s rule2

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
simpson s rule3
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
using data
Using Data
  • Given table of data, use trapezoidal rule to determine area under the curve
    • dx = ?
using data1
Using Data
  • Given table of data, use Simpson's rule to determine area under the curve
assignment
Assignment
  • Lesson 6.5
  • Page 250
  • Exercises 1 – 21 odd