MATH 2B Lecture Notes - Lecture 3: Riemann Sum

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 2B
• Professor: Michiel F Kosters

Document Summary

Math 2b lecture 3 5. 2: the definite integral. Definition: if f is a function defined for a < x < b, divide [a, b] into n subintervals of same width. =(cid:3029) (cid:3028) , let (cid:2868), (cid:2869), be the endpoints of subintervals, and let be a sample point in. [ (cid:2869),], then the definite integral of f from a to b is (cid:1858)(cid:4666)(cid:4667)(cid:1856) Lim (cid:1858)(cid:4666) (cid:4667) provided it exists and does not depend on choice of . If limit exists, we say f is integrable on. Notation: a = lower bound, b = upper bound f(x) = integrand. Theorem: if f is continuous on [a, b] or f has only a finite number of jump points, then f is integrable on [a, b], that is (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:3029)(cid:3028) exists. (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:3029)(cid:3028) (cid:3028)(cid:3029: (cid:1858)(cid:4666)(cid:4667)(cid:1856)=(cid:882) (cid:3028)(cid:3028, (cid:1855)(cid:1856)=(cid:1855)(cid:4666)(cid:1854) (cid:1853)(cid:4667) (cid:3029)(cid:3028, ((cid:1858)(cid:4666)(cid:4667)+(cid:1859)(cid:4666)(cid:4667))(cid:1856) (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:1859)(cid:4666)(cid:4667)(cid:1856) (cid:3029)(cid:3028) (cid:3029)(cid:3028) (cid:3029)(cid:3028: (cid:1855)(cid:1858)(cid:4666)(cid:4667)(cid:1856)= (cid:1855) (cid:1858)(cid:4666)(cid:4667)(cid:1856) (cid:3029)(cid:3028) (cid:3029)(cid:3028, (cid:1858)(cid:4666)(cid:4667)(cid:1856)