advertisement

Math 142 Lecture Notes for Section Section 6.4 - 1 The Definite Integral Definition 6.4.1: If f is a function defined for a ≤ x ≤ b, we can divide the interval [a, b] into n subintervals of equal width ∆x = (b − a)/n. We let x0 = a, xn = b and x∗1 , x∗2 , . . . , x∗n be any sample points in the subintervals, so x∗i lies in the ith subinterval [xi−1 , xi ]. Then the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. Example 6.4.2: R3 Approximate 0 (2x2 − x − 2)dx by using the Riemann sum with 6 subintervals, taking the sample points to be the midpoints of each subinterval. Math 142 Lecture Notes for Section Example 6.4.3: (a) R3 (b) R4 (c) R 1 2dx 0 |x − 2|dx ( −3)3 4x + 5dx 2 Math 142 Lecture Notes for Section 3 Order Properties of Definite Integrals: Assume that all integrals given below exist and that a ≤ b. (1) If f (x) ≥ 0, then (2) If f (x) ≤ g(x) on a ≤ x ≤ b, then (3) if m ≤ f (x) ≤ M on a ≤ x ≤ b, then Example 6.4.4: Use the bounds of the function y = x3 to estimate Z 4 x3 dx 0 Section 6.4 Suggested Homework 1-11 (odd, n = 10 only), 23-29(odd), 33-41(odd), 67