RAC integration theorems

Question 1

relation of upper and lower integrals

Answer 1

if f:[a,b]->ℝ then lower integral ≤ upper integral

Question 2

sums of different partitions (Lemma)

Answer 2

Suppose that P and Q are both partitions of the same interval. If P⊆Q, then
L(f,P)≤L(f,Q) and U(f,P)≥U(f,Q)

Question 3

ε-p definiton of integration

Answer 3

f: [a,b]->[0,∞) where f is bounded and non-negative is integrable if and only if:
For each ε>0, there exists a partition of P such that U(f,p)-L(f,p)

Question 4

monotonicity and integrability

Answer 4

if f:[a,b]->ℝ is monotonic on [a,b], then f is integrable on [a,b]

Question 5

differentiability, continuity and integrability

Answer 5

The following implications hold for f:[a,b]->ℝ and c∈(a,b):
1. if f is differentiable at c, then f is continuous at c
2. if f is continuous on [a,b], then f is integrable on [a,b]
In general, the reverse implications DO NOT hold.

Question 6

splitting an integral at limits

Answer 6

if a less than c less than b are all real.

if f is integrable on [a,c] and [c,b] then f is integrable on [a,b] with ∫ᵇₐf = ∫ᶜₐf + ∫ᵇ꜀f

Question 7

Linearity theorem

Answer 7

if α∈ℝ and f,g are integrable functions on [a,b], then f+g , αf and fg are all integrable functions on [a,b] and the following identities hold:

(a) ∫ᵇₐ(f+g) = ∫ᵇₐf + ∫ᵇₐg
(b) ∫ᵇₐ(αf) = α ∫ᵇₐf

Question 8

bounds of the integral

Answer 8

if f is integrable on [a,b] and m≤f(x)≤M for all x∈[a,b], then m(b-a)≤∫ᵇₐf≤M(b-a)

Question 9

Second fundamental theorem of calculus

Answer 9

if f is integrable on [a,b] anf f=g’ for some (differentiable) function g on [a,b], then ∫ᵇₐf = g(b) - g(a)
Any such function g is called an anti derivative of f, and these are unique up to additive constants C∈ℝ, so we adopt the notation ∫f:= g + C

Question 10

First fundamental Theorem of calculus

Answer 10

If f is integrable on [a,b] and continuous at c∈[a,b], then the function F is defined by
F(x) := ∫ˣₐf, if x∈[a,b],
and F(x) is differentiable at c and F'(c)=f(c)

Question 11

Integration by parts

Answer 11

If f and g are differentiable functions on [a,b] and both derivative functions f’ and g’ are integrable on [a,b], then
∫ᵇₐfg’ = [fg]ᵇₐ - ∫ᵇₐf’g

Question 12

Axiom of completeness

Answer 12

if X is non-empty and has an upperbound then supX exists (same for lower bound/inf)

Question 13

Partition Lemma

Answer 13

if P⊆Q are partitions of [a,b] then L(f,P)≤L(f,Q) and U(f,P)≥U(f,Q)

Question 14

Corollary of Partition Lemma

Answer 14

If R and S are partitions of [a,b] then:

L(f,R)≤U(f,S)

Question 15

Continuity and integrability

Answer 15

if f: [a,b]->[0,∞) is continuous, the f is integrable

Question 16

Theorem integrable/no integrable functions

Answer 16

if f: [a,b]->[0,∞), then:
lower integral ≤ upper integral
There exists f such that:
lower integral < upper integral

Question 17

monotonicty theorem

Answer 17

if f: [a,b]->[0,∞) is monotonic, then f is integrable

Question 18

continuity thoerem

Answer 18

if f: [a,b]->[0,∞) is continuous, then f is integrable

Question 19

sum of integrals theorem

Answer 19

Given a

Question 20

multiplication of integrals theorem

Answer 20

if f,g: [a,b]->ℝ are bounded and integrable, then fg: [a,b] -> ℝ is also integrable

Question 21

theorem substitution

Answer 21

if g: [a,b]->ℝ is differentiable and f:[a,b]->ℝ is continuous on g([a,b]) = [g(a),g(b)], then ∫ᵇₐf(g(x))g’(x)dx = ∫ᵍ⁽ᵇ⁾₉₍ₐ₎f(u)du

Question 22

Comparison test for integrals

Answer 22

if f and g are integrable on [a,b] for all b>a and 0≤g(x)≤f(x) on [a,∞) then:

(i) if ∫∞ₐ f converges => ∫∞ₐ converges
(ii) if ∫∞ₐ g diverges => ∫∞ₐ f diverges

Question 23

Solution to first order DE

Answer 23

if f:[a,b]->ℝ is continuous, the the initial value from has the solution
y(x) = ∫ˣₐ + y(0)