5. The Riemann Integral

Question 1

Definition
Partition

Question 2

Definition
m_{i}, M_{i}

Question 3

Definition
Upper and Lower Sums

Question 4

Lemma
Upper Sum >= Lower Sum
Proof

Question 5

Definition
Upper and Lower Integrals

Question 6

Definition
The Riemann Integral

Question 7

Definition
Refinements

Question 8

Lemma
More Refined Partitions are Better
Proof

Question 9

Lemma
Upper Sum >= Lower Sum (Partitions)
Proof

Question 10

Corollary
Upper Integral >= Lower Integral

Question 11

Lemma
Integrability Condition
Proof

Question 12

Theorem
Uniform Continuity

Question 13

Theorem
Integrability of Continuous Functions
Proof

Question 14

Theorem
Integrability of Monotone Functions
Proof

Question 15

Theorem
Small Mesh

Question 16

\int a f(x) dx, \int f(x) + g(x) dx

Question 17

Lemma
sup(f+g) = sum(f) + sum(g)
Proof

Question 18

Theorem
Linearity of the Integral

Question 19

Theorem
Monotonicity of the Integral

Question 20

Theorem
Continuous Functions of an Integrable Function

Question 21

Corollary
Integrability of Products

Question 22

Corollary
The Triangle Inequality
Proof

Question 23

Corollary
Addition of Ranges

Question 24

Theorem
FTC I
Proof

Question 25

Theorem
FTC II
Proof

Question 26

Theorem
Integration by Parts
Proof

Question 27

Theorem
Integration by Substitution
Proof

Question 28

Definition
Improper Integrals

Question 29

Theorem
Comparison Test for Improper Integrals
Proof