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Below is an alternative but equivalent definition of the definite integral of f: R R between a and b, given x = a+k(4) for 0 sk

Below is an alternative but equivalent definition of the definite integral of f: R R between a and b, given x = a+k(4) for 0 sk

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 7E: For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={...
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Below is an alternative but equivalent definition of the definite integral of f: R → R between a and b, given x = a +k (4) for 0 <k<n: n-1 b-a I„f,a, b) = | f(*) k=0 f (x)dx: = lim I„(f,a,b) i) For f(x)= x2 , calculate I5(f,-2,8). Show your working. ii) We now define b -a Inf,a,b) =| k=1 For f(x)= x² , calculate J5(f,-2,8). Show your working. iv) We call a function f: R →R monotone increasing if x> y implies f(x) > f (y). Show that, if f is monotone increasing, then, for any a,be R,a < b , and ne N I„(f, a,b)< Jn(f ,a, b) v) Without formally proving it, explain why: lim J„f , a, b) = f (x)dx = lim If,a,b) 1 00
Transcribed Image Text:Below is an alternative but equivalent definition of the definite integral of f: R → R between a and b, given x = a +k (4) for 0 <k<n: n-1 b-a I„f,a, b) = | f(*) k=0 f (x)dx: = lim I„(f,a,b) i) For f(x)= x2 , calculate I5(f,-2,8). Show your working. ii) We now define b -a Inf,a,b) =| k=1 For f(x)= x² , calculate J5(f,-2,8). Show your working. iv) We call a function f: R →R monotone increasing if x> y implies f(x) > f (y). Show that, if f is monotone increasing, then, for any a,be R,a < b , and ne N I„(f, a,b)< Jn(f ,a, b) v) Without formally proving it, explain why: lim J„f , a, b) = f (x)dx = lim If,a,b) 1 00
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