Integration on a segment

Learning tasks

A) Uniform continuity
1) Establishing uniform continuity from the definition	Ex 1 Ex 2 Ex 3
2) Disproving uniform continuity	Ex 4 Ex 5 Ex 6
3) Using Heine's theorem in practice	Ex 7 Ex 8 Ex 9
B) Step and piecewise continuous functions
4) Identifying step and piecewise continuous functions	Ex 10 Ex 11 Ex 12
5) Constructing subdivisions adapted to a function	Ex 13 Ex 14 Ex 15
6) Using the sub-algebra structure of piecewise continuous functions	Ex 16 Ex 17 Ex 18
7) Approximating a continuous function by step functions	The three exercises in this subsection re-derive the construction of T2.1 (Uniform approximation by step functions) in concrete cases: a linear \(f\), a general continuous \(f\) on \([0, 1]\), and a sign-preserving variant. The proof technique --- Heine + uniform partition + sampling at left endpoints --- is exactly the proof of T2.1 in the continuous case. Ex 19 Ex 20 Ex 21
C) Integral of a piecewise continuous function
8) Computing integrals using parity\(\virgule\) periodicity\(\virgule\) Chasles	Ex 22 Ex 23 Ex 24
9) Establishing inequalities via positivity and monotonicity	Ex 25 Ex 26 Ex 27
10) Computing the mean value of a piecewise continuous function	Ex 28 Ex 29 Ex 30
D) Riemann sums
11) Recognizing a sum as a Riemann sum	Ex 31 Ex 32 Ex 33
12) Computing limits by integral identification	Ex 34 Ex 35 Ex 36
13) Using Riemann sums to bound a sequence	Ex 37 Ex 38 Ex 39
E) Link with primitives
14) Differentiating an integral with a moving upper bound	Ex 40 Ex 41 Ex 42
15) Applying integration by parts	Ex 43 Ex 44 Ex 45
16) Applying change of variable	Ex 46 Ex 47 Ex 48
F) Global Taylor formulas
17) Writing Taylor with integral remainder	Ex 49 Ex 50 Ex 51
18) Bounding a function via Taylor-Lagrange inequality	Ex 52 Ex 53 Ex 54
19) Proving inequalities via Taylor	Ex 55 Ex 56 Ex 57

Lesson
Text book

Exercises Correction
A) Uniform continuity
    1) Establishing uniform continuity from the definition Ex 1 Ex 2 Ex 3
    2) Disproving uniform continuity Ex 4 Ex 5 Ex 6
    3) Using Heine's theorem in practice Ex 7 Ex 8 Ex 9
B) Step and piecewise continuous functions
    4) Identifying step and piecewise continuous functions Ex 10 Ex 11 Ex 12
    5) Constructing subdivisions adapted to a function Ex 13 Ex 14 Ex 15
    6) Using the sub-algebra structure of piecewise continuous functions Ex 16 Ex 17 Ex 18
    7) Approximating a continuous function by step functions

The three exercises in this subsection re-derive the construction of T2.1 (Uniform approximation by step functions) in concrete cases: a linear \(f\), a general continuous \(f\) on \([0, 1]\), and a sign-preserving variant. The proof technique --- Heine + uniform partition + sampling at left endpoints --- is exactly the proof of T2.1 in the continuous case.

Ex 19 Ex 20 Ex 21
C) Integral of a piecewise continuous function
    8) Computing integrals using parity\(\virgule\) periodicity\(\virgule\) Chasles Ex 22 Ex 23 Ex 24
    9) Establishing inequalities via positivity and monotonicity Ex 25 Ex 26 Ex 27
    10) Computing the mean value of a piecewise continuous function Ex 28 Ex 29 Ex 30
D) Riemann sums
    11) Recognizing a sum as a Riemann sum Ex 31 Ex 32 Ex 33
    12) Computing limits by integral identification Ex 34 Ex 35 Ex 36
    13) Using Riemann sums to bound a sequence Ex 37 Ex 38 Ex 39
E) Link with primitives
    14) Differentiating an integral with a moving upper bound Ex 40 Ex 41 Ex 42
    15) Applying integration by parts Ex 43 Ex 44 Ex 45
    16) Applying change of variable Ex 46 Ex 47 Ex 48
F) Global Taylor formulas
    17) Writing Taylor with integral remainder Ex 49 Ex 50 Ex 51
    18) Bounding a function via Taylor-Lagrange inequality Ex 52 Ex 53 Ex 54
    19) Proving inequalities via Taylor Ex 55 Ex 56 Ex 57

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