MATH 1920 Lecture Notes - Lecture 18: Riemann Sum, Multiple Integral, Joule

• School: Cornell University
• Department: Mathematics
• Course: MATH 1920
• Professor: Andrew Sale

Document Summary

Function f(x, y) on a rectangle r = [a, b] x [c, d] Aim: (net) volume = r f(x, y) da d = ym yj c = y0 a = x0 xi. Partition r into smaller rectangles: a = x0 < x1 < < xn = b c = y0 < y1 < < ym = d. X = xi - xi - 1 = #$% Y = yj - yj - 1 = ($) Definition: the double integral of f(x, y) over r is given by the double riemann sum: We say f is integrable over r if the limit exists. Theorem: if f(x, y) is continuous on a rectangle r = [a, b] x [c, d] then f is integrable over r. Fix x in [a, b] and take a slice out of the solid region. Area od the slice: ()= (,) (net) volume = ()