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Applied Mathematics
Advanced Calculus and
Methods of Mathematical
Physics
Lecture 2 – Riemann Integral
Taylor Series
Notes

Lecturer: Soren PetratDate of lecture:
February 4, 2022
Author: Lirik Maxhuni

Riemann Integral
Let 𝑓[𝑎, 𝑏] → 𝑅 be bounded

Firstly, let’s define what is a partition P of [𝑎, 𝑏] is a finite set of points
𝑥0 ,
...
There are two Riemann Sums
...
Their difference stands in, that the URS
overapproximates the area of under the curve, whereas the LRS
underapproximates it
...


To visualize the difference between supremum and infimum, we have
illustrated them below
...

Upper Riemann Sum logical definition
Given the area under the curve is always smaller than the area of the
rectangles, make the rectangles as small as possible
...

𝑏

𝑏

𝑏
̅̅̅̅̅

𝑏

𝑏

∫ 𝑓 ≡ ∫ 𝑓(𝑥)𝑑𝑥 (≡ ∫ 𝑑𝑥 𝑓(𝑥)) ∶= ∫ 𝑓 = ∫ 𝑓
𝑎

𝑎

𝑎

𝑎

𝑎

Non-Integrable functions
However, there exist functions that are not Riemann integrable
...
:
1,
𝑓(𝑥) = {
0,

𝑥∈𝑄
𝑒𝑙𝑠𝑒

Then we would have:
1
̅̅̅̅̅

∫ 𝑓 (𝑥)𝑑𝑥 = 1
0
1

∫ 𝑓 (𝑥)𝑑𝑥 = 0
0

As we can see, the lower and upper Riemann Integrals do not match, therefore
the function is not Reimann integrable
...
𝑓 is differentiable at some 𝑡 ⇒ 𝑓 is continuous at 𝑡
2
...
Mean Value Theorem: let 𝑓 [𝑎, 𝑏] → 𝑅 be continuous and differentiable,
then there is:
𝑧 ∈ (𝑎, 𝑏) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
𝑓 ′(𝑧) =

𝑓(𝑏) − 𝑓(𝑎)
𝑏−𝑎

4
...
Fundamental Theorem of Calculus
Version 1: let 𝑓 be integrable on [𝑎, 𝑏], F continuous on [𝑎, 𝑏] and
differentiable on [𝑎, 𝑏], with 𝐹′(𝑥) = 𝑓(𝑥), ∀ 𝑥 ∈ (𝑎, 𝑏)
...

Define 𝐹[𝑎, 𝑏] → 𝑅 by
𝑥

𝐹(𝑥) − 𝐹(𝑎) = ∫ 𝑓(𝑡)𝑑𝑡
𝑎

Then 𝐹 is continuous on [𝑎, 𝑏] and diffable in 𝑡
...
𝐼 → 𝑅 is of class 𝐶 𝑛 (𝐼) if 𝑓 is 𝑛 times differentiable and 𝑓, 𝑓 ′, ⋯ , 𝑓 (𝑛) are
continuous
...

Note: We know that lim

𝑅𝑛,𝑐 (𝑥)

𝑥→𝑐 (𝑥−𝑐)𝑛

=0



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