4.4 the definite integral by flu11339

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									4.4 The Definite Integral                              1/8




Recall:
     n
                                      n           
A ≈ ∑f (ci ) ∆xi            A = lim  ∑f (ci ) ∆xi 
    i =1
                                n →∞
                                      i =1        
4.4 The Definite Integral                                  2/8
                        For any function f defined on
                        [a , b ] , the definite integral of f
                        from a to b is
                          b                  n            
                         ∫a f (x )dx = nlim  ∑f (ci ) ∆xi  ,
                                         →∞
                                             i =1         
                        whenever the limit exists and is
                        the same for any choice of
evaluation points, c1 , c2 ,…, cn . When the limit exists,
we say that f is integrable on [a , b ] .
4.4 The Definite Integral                       3/8
           1  4
        ∫2  2 x dx
Evaluate        
geometrically.
      4 y

      3

      2

      1
                                            x
 -1         1   2   3   4   5   6   7   8
      -1
4.4 The Definite Integral                     4/8
               3

Evaluate   ∫
           −3
                   9 − x 2dx geometrically.
4.4 The Definite Integral                               5/8
     y

 4
                                       If A 1 = 4.7467 and
 3
                                       A 2 = 5.67 , find
                                        5
                                       ∫ f (x )dx
 2
         A1           A2                0
 1
                                   x
          1   2   3        4   5   6


Generalize:
4.4 The Definite Integral                        6/8
Write as a single integral:
            4               9

1.      ∫1
                f (x )dx + ∫ f (x )dx
                           4


            7               7

2.      ∫3
                f (x )dx − ∫ f (x )dx
                           5



Sketch the area corresponding to the integral.
    π
∫0
        sin xdx
4.4 The Definite Integral            7/8
               π
Evaluate   ∫
           0
                   cos xdx



     y




                                 x
                    π/2      π
4.4 The Definite Integral                                                   8/8
#9 from the 1998 exam:




The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the
graph shown above. Of the following, which best approximates the total number
of barrels of oil that passed through the pipeline that day?


(A) 500 (B) 600        (C) 2,400       (D) 3,000       (E) 4,800

AB – 67% correct       BC – 80% correct

								
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