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
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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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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