Riemann Integral

Question 1

When is a function f bounded on [a,b]

Answer 1

A function f is bounded on [a,b] if :

There exists M >= 0 s.t for all x in [a,b] mod(f(x)) <= M

Question 2

What is a subdivision P of [a,b]

Answer 2

Subdivision P of [a,b] is a collection of points in [a,b] such a,b is in P
P = {x1,…,Xn} a= X0 < X1 < … < Xn = b
(split into n sub intervals)

Question 3

What is lower Riemann sum

Answer 3

Lower Riemann sum is:
L(f,P) = sum i = 1 to n (Xi - X(i-1)) inf f [in interval [X(i-1), Xi]
Xi are intervals of subdivision P, if is a bounded function [a,b]

Question 4

What is upper Riemann sum

Answer 4

Upper Riemann sum is:

U(f,p) = sum i =1 to n mod(Xi - X(i-1)) sup f [in interval [X(i-1), Xi]]

Question 5

What are definitions of in f and sup f

Answer 5

Definitions of inf f and sup f are:
Inf f = inf{f(x) | X E D} = smallest value of f over D
Sup f = sup{f(x) | X E D } = largest value of f over D

Question 6

What is relationship between lower and upper sum of f and extension

Answer 6

Relationship between lower and upper sim of f is:
L(f,P) <= U(f,P) for f a bounded function and P a subdivision of [a,b]
Extension is
L(f,P) <= U(f,Q) where Q is another subdivision of x

Question 7

If Q,P are subdivisions of [a,b] then what is a refinement

Answer 7

If Q,P are subdivisions of P, then:

Q is finer or a refinement of P if Q >= P

Question 8

What is definition of upper Riemann integral

Answer 8

Definition of upper Riemann integral is:
Integral b(bar) down to a f = inf{ U(f,P| P is a subdivision of [a,b]}

Question 9

What is definition of lower integral of f

Answer 9

Definition of lower integral of f is:

Integral b down to a(bar) f = sup{ L(F,P)| P is subdivision of [a,b]

Question 10

When is a function f Riemann integrable on [a,b]

Answer 10

A function is Riemann integrable on [a,b] if :

Integral b down to a(bar) = integral b(bar) down to a = integral a down to b

Question 11

What is Cauchy criterion of integrability

Answer 11

Cauchy criterion of integrability is:
If f : [a,b] to R is bounded then f is integrable iff
Exists P subdivision of [a,b] s.t U(f,P) - L(f,P) < epsilon