Integration

Question 1

Lemme 13.1

Let A and B be bounded sets. If A is contained in B what can be said about infA and infB? What can S said about supA and supB?

Answer 1

infB is less than or equal to infA

and

supA is less than or equal to supB

Question 2

Lemme 13.2

Let A be a bounded set and
let B =  {cx: x is in A and c is greater than 0}.

What can be said about infA and infB?
What can be said about supA and supB?

Answer 2

cinfA = infB

and

csupA = supB

Question 3

Lemme 13.3

Let A be a bounded set and
Let B = {cx : x is in A and c is less than 0}.

What can be said about infA and infB?

What can be said about supA and supB?

Answer 3

cinfA = supB

and

csupA = infB

Question 4

What is the definition of a Partition of the interval [a,b]?

Answer 4

A partition of the interval [a, b] is a finite collection of point is [a, b], one of which is A and one of which is b.

Question 5

Suppose f is bounded on [a, b] and P = {t(0), t(1), … , t(n) } is a partition on [a, b].

Let m(i) = inf{ f(x): x is in [ t(i - 1), t(i) ] }

and

Let M(i) = sup{ f(x): x is in [ t(i - 1), t(i) ] }

What are the lower and upper sums of f on P?

Answer 5

The lower sum of f on P is:

L(f, P) = sum( m(i)*( t(i - 1) - t(i) ), 1, n )

The upper sum of f on P is:

U(f, P) = sum( M(i)*( t(i - 1) - t(i) ), 1, n )

Question 6

Lemma 13.4

Let P and Q be partitions on [a, b] and suppose P is contained in Q.

What can be said about L(f, P) and L(f, Q)?

What can be said about U(f, P) and U(f, Q)?

Answer 6

L(f, P) is less than or equal to
L(f, Q)

and

U(f, P) is greater than or equal to
U(f, Q)

Question 7

Theorem 1

Let P and Q be partitions on [a, b] and let f be a bounded function on [a, b].

What can be said about L(f, P) and U(f, Q)?

Answer 7

L(f, P) is less than or equal to

U(f, Q).

Question 8

Let A = { L(f, P): P is a part on [a,b] }

and

let B = { U(f, P): P is a part on [a, b]}.

What can be said about supA and infB ?

Answer 8

supA is less than or equal to infB.

In fact we have that for all P, where P is a partition on [a, b],

L(f, P) <= supA <=infB <= U(f, P)

Question 9

Let A = { L(f, P): P is a part on [a,b] }

and

let B = { U(f, P): P is a part on [a, b]}.

Suppose supA = a = infB. What can be aid about a?

Answer 9

If supA = a = infB then a is the only number with this property

Question 10

Definition :

A function f which is bounded on [a, b] is integrable on [a, b] if and only if …

Answer 10

supA = infB.

In this case the common number is called the integral of f on [a, b]. It is denoted
int(f, a, b)

int(f, a, b) is all called the area of R(f, a, b) if
f(x) >= for all x in [a, b].

For all partitions P on [a, b] we have that:
L(f, P) <= int(f, a, b) <= U(f, P).

Question 11

Theorem 2 ( Alternative definition for integrable)

If f is bounded on [a, b] then f is integrable on [a, b] if and only if…

Answer 11

For all epsilon greater than 0 there is a partition, P, on [a, b] such that

U(f, P) - L(f, P) is less than epsilon.

Question 12

Theorem 3

If f is continuous on [a, b] then f is integrable on [a, b]

Answer 12

The proof is this uses the fact that is f is continuous on [a, b] then f is bounded on [a,b] and uniformly continuous on [a, b].

Question 13

Theorem 4 (a)

Let a < c < b. f is integrable on [a, b] if and only if f is integrable on [a, c] and [c, b].

Answer 13

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Question 14

Theorem 4(b)

Let a < c < b. If f is integrable on [a, b] then:

int(f, a, b) = int(f, a, c) + int(f, c, b)

Answer 14

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Question 15

Theorem 5:

If f and g are integrable on [a, b] then f + g is integrable on [a, b] and

int(f+ g, a, b) = int(f, a, b) + int(g, a, b)

Answer 15

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Question 16

Theorem 6:

If f is integrable on [a, b] then cf is integrable on [a, b] and

int(cf, a, b) = c*int(f, a, b)

Answer 16

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Question 17

If f is integrable on [a, b] and
m <= f(x) <= M for all x then

m(b -a) < int(f, a, b) < M(b - a)

Answer 17

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Question 18

If f is integrable on [a, b] and F is defined on [a, b] by

F(x) = int( f(t), a, x)

Then F is continuous on [a, b]

Answer 18

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