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Unformatted text preview: 1 5.3 The Definite Integral We have approximated the area under the graph of a function by using rect- angles. The thinner the rectangles the better the approximation. Exact area under the graph can be defined as a limit of the approximations , where x is the ( ) of each rectangle and f ( x i ) is the of the i th rectangle. This motivates the following defintion: Definition If f is a function on the interval [ a,b ], we divide the [ a,b ] into n subintervals of equal width x = . The of f from a to b is: = where x k is point inside k th subinterval. REMARKS: The definite integral is a It represents the area under the graph only if on [ a,b ] If f ( x ) < 0 somewhere inside the interval, the definite integral represent the area under the graph. If f ( x ) < 0 somewhere on [ a,b ] the definite integral represents the sum of the area under the graph where MINUS the area between the axis and the graph where ....
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