13112an5.3-5 - c Kendra Kilmer March 1 2012 Section 5.2 The...

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c circlecopyrt Kendra Kilmer March 1, 2012 Section 5.2 - The Definite Integral Definition of a Definite Integral: If f is a function defined for a x b , we divide the interval [ a , b ] into n subintervals of equal width x =( b a ) / n . We let x 0 = a , x n = b , and x 1 , x 2 ,..., x n be any sample points in these subintervals, so x i lies in the i th subinterval [ x i 1 , x i ] . Then the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that f is integrable on [ a , b ] . Note: Geometrically, the definite integral represents the cumulative sum of the signed areas between the graph of f ( x ) and the x -axis from x = a to x = b , where areas above the x -axis are counted positively and areas below the x -axis are counted negatively. Example 1: Approximate integraldisplay 3 0 ( 2 x 2 x 2 ) dx by using the Riemann sum with 6 equal subintervals, taking the sample points to be the midpoints of each subinterval. 3
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c circlecopyrt Kendra Kilmer March 1, 2012 Example 2: Calculate the following given f ( x ) below g h B C f(x) Area of A is 0.3 Area of B is 2.5 Area of C is 2 Area of D is 0.75 j D e A a) integraldisplay 0 g f ( x ) dx
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