lecture31a - Lecture 31 (Chapter 5, Sec. 31): Area and...

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Unformatted text preview: Lecture 31 (Chapter 5, Sec. 31): Area and Distance 6- ? How to flnd the area of the region that lies under the curve y = f ( x ) from a to b ? ex. Let f ( x ) = p x + 1 and consider the area be- neath the graph of the function on [0 ; 1]. 2 Let R n be the sum of the areas of n rectangles with equal width and height 1) Find R 4 ( n = 4): 2) Find R 8 ( n = 8): 3 For any n , R n = As n ! 1 , what happens to our approximation? We deflne the area as lim n !1 R n = lim n !1 1 n p x 1 + 1 + ::: + p x i + 1 + ::: + p x n + 1 / In general, To flnd the area under the curve y = f ( x ) on [ a;b ] : Divide [ a;b ] into n subintervals using partition a = = b This creates n subintervals: 4 Then consider n rectangles, one for each subinterval: Width x = Height: f ( x / i ), where x / i is Area A can be approximated by the sum of the areas of the n rectangles: This sum is called a Riemann sum . We use sigma notation to express the sum more con- cisely as A Generally, if f is continuous, as the number of subin-...
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This note was uploaded on 05/17/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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lecture31a - Lecture 31 (Chapter 5, Sec. 31): Area and...

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