Approximate Integration

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					      7.7 Approximate Integration
If we wish to evaluate a definite integral involving a
function whose antiderivative we cannot find, then we
must resort to an approximation technique. We
describe three such techniques in this section.

 A. Midpoint Rule
 B. Trapezoidal Rule
 C. Simpson’s Rule
                A. Midpoint Rule                                                                                     y

                                                                                                                         y  f (x)
        Let f be continuous on [a, b]. The Midpoint Rule for
                        b
        approximating a f ( x)dx is given by

        
        a
            b
            f ( x)dx  M n 
                               ba
                                n
                                                                          
                                   f ( x1 )  f ( x 2 )  ...  f ( x n ) where x i  x i 1  x i 
                                                                                     1
                                                                                     2
                                                                                                                                                                          x
                                                                                                                         a x1        x2        …           xn b


                B. Trapezoidal Rule                                                                                      y

                                                                                                                             y  f (x)

        Let f be continuous on [a, b]. The Trapezoidal Rule for
        approximating bf ( x)dx is given by
                                      a
                          ba
                               f ( x0 )  2 f ( x1 )  2 f ( x 2 )  ...  2 f x n1   f ( x n )
            b
        
        a
          f ( x)dx  Tn 
                           2n
                                                                                                                                                                              x
                                                                                                                         a  x0      x1   x2     …           xn  b

                C. Simpson’s Rule (n is even)                                                                                                  y  f (x)
                                                                                                                         y
        Let f be continuous on [a, b]. The Simpson’s Rule for
                         b
        approximating a f ( x)dx is given by
                       ba
                            f ( x0 )  4 f ( x1 )  2 f ( x 2 )  ...  2 f x n2   4 f ( x n1 )  f ( x n )
    b

a
    f ( x)dx  S n 
                        3n
                                                                                                                                                                      x
                                                                                                                         a  x0            …                xn  b
                                                       Example 1:
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) the Simpson’s Rule
to approximate the given integral with the specified value of n. (Round your
answers to six decimal places.)
                                                  3
                                              
                                                        1
                                                           dx, n  10
                                               2      ln x

   (a) Trapezoidal Rule
                     f (2)  2 f (2.1)  2 f (2.2)  2 f (2.3)  ...  2 f (2.9)  f (3)
 3 1              1
2     dx  T10 
        ln x                   2 10
                                 1.119061
        (b) Midpoint Rule
                            dx  M 10   f (2.05)  f (2.15)  f (2.25)  f (2.35)  ...  f (2.85)  f 2.95)
                   3
               
                         1               1
               2       ln x             10
                                       1.118107

        (c) Simpson’s Rule
         1                1
                               f (2)  4 f (2.1)  2 f (2.2)  4 f (2.3)  ...  2 f (2.8)  4 f (2.9)  f (3)
    3
2      ln x
             dx  S10 
                        3 10
                         1.118428
                                       Example 2:
 The width (in meters) of a kidney-shaped swimming pool
 were measured at 2-meter intervals as indicated in the
 figure. Use Simpson’s Rule to estimate the area of the pool.

                                      6.8     5.6     5.0
                               7.2                            4.8
                     6.2                                             4.8



Solutions:
                     Let x = distance from the left end of the pool
                        w = w(x) = width at x
                                 b  a 16  0
                      n  8,                2
                                   n     8

                                  10  46.2  27.2  ...  24.8  4(4.8)  10
         16
         
                                2
Area        w( x)dx  S 8 
         0                      3
                                 84 m 2

				
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