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Unformatted text preview: Lecture 25 — The Definite Integral If f is continuous on [ a,b ] , partition [ a,b ] into n subintervals of equal width Δ x i = and let x * i be any point in subinterval i : [ x i 1 ,x i ] . Then the definite integral from a to b is written: Z b a f ( x ) d x = When f ( x ) is nonnegative, the definite integral will represent: But, it has a new meaning when f takes negative values: Notation: Z b a f ( x ) d x The process of evaluating the definite integral is called: If it is not advantageous to space the partition equally, or if f is not continuous, what then? A function f is said to be integrable on interval [ a,b ] if, independent of the choice of partition points and x * i in each subinterval: This will be true for functions that are continuous or Express lim n →∞ n X i =1 x * i e x * i 3 Δ x i as a definite integral on [ a,b ] . Application: A 100foot cable of uniform density and weighing 2 lbs/ft is suspended from the roof of a tall building. Express the amount of work that isa tall building....
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This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
 Spring '08
 ALL
 Calculus

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