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Definite Integral

Problems

Definition of the Definite Integral

Properties of the Definite Integral

Problem : Compute U 3(f, 0, 3) and L 3(f, 0, 3) for f (x) = (x - 2)2 .

We subdivide the interval [0, 3] into the three intervals [0, 1] , [1, 2] , [2, 3] , so that M 1 = f (0) = 4 , M 2 = f (1) = 1 , M 3 = f (3) = 1 , and m 1 = f (1) = 1 , m 2 = f (2) = 0 , m 3 = f (2) = 0 . Therefore,


U 3(f, 0, 3)   = M i = (4 + 1 + 1) = 2  
L 3(f, 0, 3)   = m i = (1 + 0 + 0) =  

We may conclude that (x - 2)2 dx≤2 .

Problem : Compute - 1dx .

This definite integral is equal to the area of a rectangle with height 1 unit and length (b - a) units lying below the x -axis. The area therefore counts as negative, so the definite integral equals - (1)(b - a) = a - b .

Problem : Compute xdx .

Since R(x, 0, 2) is a triangle with base of length 2 and a height of 2 , we know that the area should be (2)(2) = 2 . We check that this agrees with the Riemann sum definition:


U n(x, 0, 2) =  
  = i  
  =  
  = 2 -  
  = 2  

and similarly for L n(x, 0, 2) .

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