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# L25 - Lecture 25 The Denite Integral If f is continuous on...

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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:

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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 100 -foot cable of uniform density and weighing 2 lbs/ft is suspended from the roof of a tall building. Express the amount of work that is required to raise the cable to the top of the building as a definite integral by using a Riemann Sum.

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How can we evaluate integrals at this point?
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