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4.4 The Definite Integral 18

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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
                                            n             
                         ∫a f (x)dx = lim  ∑ f ( ci ) ∆xi  ,
                           b

                                       n→∞
                                            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 geometrically.
Evaluate  

      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 2 dx geometrically.
4.4 The Definite Integral                                    5/8
     y

 4
                                       If A 1 = 4.7467 and
 3                                      A 2 = 5.67 , find
                                           5
                                       ∫
 2
         A1           A2                       f (x)dx
                                        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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posted:7/30/2011
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