L24 - Lecture 24 Area and Distance Accumulated Change...

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Lecture 24 — Area and Distance, Accumulated Change, Riemann Sums We used derivatives to solve problems involving rates of change. The second application of calculus is the problem of area and its interpretations. Once again, we will need limits. Find the area beneath the graph of the function f ( x ) = x 2 on the interval [ 0 , 1 ] : Let R n be the sum of the areas of n approximating rectangles each of equal width and height When n = 4 : R 4 =
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When n = 8 : R 8 = In general, R n = Looking at the graph, the approximations in this problem are closer and closer to the actual area each time, but are always By the completeness axiom, the numbers R n have which must be lim n →∞ R n = so we define the area to be
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In general, given a function f ( x ) , divide [ a,b ] into n subintervals using the partition a = = b This creates the n subintervals: on which we’ll make n rectangles, each having: width Δ x i = height f ( x * i ) , where x * i The sum of these n rectangles is called a Riemann Sum and is written: or, in sigma notation:
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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.

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L24 - Lecture 24 Area and Distance Accumulated Change...

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