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Unformatted text preview: 5.2 Definite Integrals Bernhard Reimann 30 34 k k = 2 50 ∑ 2 k k =2 50 ∑ 3 k 2 k = 2 100 ∑ n 2 + 3ν κ =1 4 ∑ 1 When we find the area under a curve by adding rectangles, the answer is called a Rieman sum . 1 2 3 1 2 3 4 2 1 1 8 V t = + subinterval partition The width of a rectangle is called a subinterval . The entire interval is called the partition . Subintervals do not all have to be the same size. → 1 2 3 1 2 3 4 2 1 1 8 V t = + subinterval partition If the partition is denoted by P , then the length of the longest subinterval is called the norm of P and is denoted by . P As gets smaller, the approximation for the area gets better. P ( 29 1 Area lim n k k P k f c x ® = = D å if P is a partition of the interval [ ] , a b → ( 29 1 lim n k k P k f c x ® = D å is called the definite integral of over . f [ ] , a b If we use subintervals of equal length, then the length of a subinterval is: b a x n ∆ = The definite integral is then given by: ( 29 1 lim n k n k f c x ® ¥ = D å → ( 29 1 lim n k n k f c x ® ¥ = D å Leibnitz introduced a simpler notation for the definite integral: ( 29 ( 29 1 lim n b k a n k f c x f x dx ® ¥ = D = å ò Note that the very small change in x becomes dx ....
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This note was uploaded on 02/16/2012 for the course CALCULUS 0064 taught by Professor Waldron during the Fall '10 term at Broward College.
 Fall '10
 waldron
 Definite Integrals, Integrals

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