MTH 252 Integral Calculus Chapter 6 Integration Section 6 5 The Definite Integral Copyright 2005 by Ron Wallace all rights reserved Partitions a x0 x1 x2 x3 xn 1 b xn A partition of a b is a collection of n 1 points Pn x0 x1 x2 x3 xn 1 xn such that a x0 x1 x2 x3 xn 1 xn b Subinterval Length kth interval xk xk xk 1 Mesh Size of a Partition max xk largest subinterval length Regular Partition xk x b a n for all k i e all of the subintervals have the same length Riemann Sum If f x is continuous over a b Pn is a partition of a b and xk is any point in the kth subinterval then n f x Dx k k k 1 is called a Riemann Sum Definite Integral n If lim max Dxk 0 f x Dx k k k 1 exists then f x is integrable and the limit is the definite integral of f x from a to b b f x dx lim a max Dxk 0 n f x Dx k 1 k k Definite Integral Equivalent simplified and more practical form Since any partition can be used use a regular partition Since xk is any point in the kth interval use xk the right endpoint Therefore n b f x dx lim f x D x k a where n k 1 b a Dx n k b a xk a k Dx a n Definite Integrals Area Problems If f x 0 over a b then under the curve If f x 0 over a b then above the curve b f x dxis the area a b f x dx is the area a Therefore geometry can be used to determine some integrals Example 2 4 x 0 2 dx p Two extensions to the definition a f x dx 0 a If a b then a b b a f x dx f x dx Properties of Definite Integrals b b a a cf x dx c f x dx b b b a a a f x g x dx f x dx g x dx b c b a a c f x dx f x dx f x dx This is true if a c b c a b or a b c