MATH 1004 Lecture Notes - Lecture 15: Riemann Integral, Trigonometric Functions, Riemann Sum

• School: Carleton University
• Department: Mathematics
• Course: MATH 1004
• Professor: Glenn Mc Rae

Document Summary

Let [a, b] be a closed interval of real numbers i. e. , [a, b] = {x : a x b}. The length of each subinterval of the form [xi, xi+1] = xi+1 xi = (cid:52)xi. Norm: |p| = length of the largest subinterval n 1(cid:88) i=0. Limit of the riemann sum as the norm of the partition p approaches to 0 is l n 1(cid:88) i=0 lim|p| 0 f (ti)(cid:52)xi = l. We say that f is riemann integrable over the interval [a, b] and l is the value of the riemann integral of f over [a, b], denoted by. L = r f (x)dx = f(b) f(a) = lim|p| 0 f (ti)(cid:52)xi. The symbols on the right side call r integral of f over [a, b]. For f (x) 0, area under f between the two lines at a and b is f (x)dx = f(b) f(a) = f(x) n 1(cid:88) i=0 (cid:105)b a (cid:90) b a (cid:90) b a.