Table of Contents
Table of Contents
“Exercises of Integral Calculus”
INTRODUCTION
DEFINITE AND INDEFINITE INTEGRALS
IMPROPER INTEGRALS
“Exercises of Integral Calculus”
“Exercises of Integral Calculus”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
definite and indefinite integrals
improper integrals
geometric applications and remarkable theorems of integral calculus.
Initial theoretical hints are also presented to make the performance of the exercises understood.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – DEFINITE AND INDEFINITE INTEGRALS
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
Exercise 33
Exercise 34
Exercise 35
Exercise 36
Exercise 37
Exercise 38
Exercise 39
Exercise 40
Exercise 41
Exercise 42
Exercise 43
Exercise 44
Exercise 45
Exercise 46
Exercise 47
Exercise 48
Exercise 49
Exercise 50
Exercise 51
Exercise 52
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II – IMPROPER INTEGRALS
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
INTRODUCTION
INTRODUCTION
In this exercise book some examples of calculations related to definite, indefinite and improper integrals are carried out.
Furthermore, the main theorems used in integral calculus are presented, as well as the remarkable geometric applications of this sector of mathematical analysis.
Integral calculus is a milestone in mathematical analysis: the search for primitives and the convergence of improper integrals have represented not only a major mathematical challenge, but also an elegant solution to many physical and applicative problems.
In order to understand in more detail what is explained in the resolution of the exercises, the theoretical context of reference is recalled at the beginning of each chapter.
What is presented in this workbook is generally addressed during the last year of scientific high schools and, above all, in the first course of mathematical analysis proposed at university level.
I
DEFINITE AND INDEFINITE INTEGRALS
DEFINITE AND INDEFINITE INTEGRALS
Considering a continuous function in a closed and bounded interval [a,b], one can define two points within any partition of the interval given by the lower bound and the upper bound as follows:
The lower and upper integral sums are constructed as follows:
We define the following quantity as an integral sum :
The limit of this integral sum (if it exists finitely) is called the Riemann integral and is indicated as follows:
E represents the convergence between the lower and upper integral sum.
The function is therefore said to be integrable in the closed interval [a,b].
A sufficient condition for integrability is given by the continuity of the function over a closed and bounded interval: a uniformly continuous function is therefore integrable.
A function is said to be absolutely integrable if its module is integrable (it goes without saying that an absolutely integrable function is integrable).
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The Riemann integral enjoys the properties of linearity,additivity and monotonicity .
In formulas we have:
Furthermore, two theorems concerning the absolute value and the integral mean hold :
The Riemann integral proposed up to now is called a definite integral and it is a functional , ie it returns a numerical value following an operation on a function of a real variable.
The geometric meaning of the integral defined according to Riemann is easy to explain.
Recalling that the upper integral sum is the area of the rectangles circumscribed to the region of the plane delimited by the graph of the function and the abscissa axis and that the lower integral sum is instead the area of the rectangles inscribed in this region , the definite integral computes exactly the area subtended between the graph of the function and the abscissa axis in the closed and bounded interval [a,b].
This result is also valid for the plane regions included between two curves, where the definite integral of the difference of the functions represents the measure of the area of that plane region (always keeping in mind that the geometric areas are positive and therefore we consider always the absolute values of the differences).
A further geometric application of the definite integral is given by the calculation of the volume and surface area of a solid of rotation . In fact, in the closed and bounded interval [a,b] the following holds:
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Instead, we call integral function (or Torricelli integral) a function given by a definite integral in which one extremum of integration is fixed while the other is variable.
The fundamental theorem of integral calculus states that given an integrable function f(x) and an integral function built on it:
Then the integral function is continuous in [a,b]. Furthermore, if f(x) is continuous, the integral function is differentiable in the open range (a,b) and holds:
The integral function is said to be primitive with respect to the integrand function.
In this way it is seen that the integral calculus represents the inverse of the differential one.
The theorem of integral calculus applied to a definite integral leads to the fundamental formula of integral calculus .
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By applying the fundamental theorem of integral calculus, the so -called indefinite integrals can be defined, i.e. operators that return functions whose derivatives coincide with the original integrand function.
Since the derivative of a constant function is null, it means that the primitive given by an indefinite integral is calculated up to an arbitrary constant, i.e. there are infinite primitives that can be grouped into families.
A sufficient condition for the existence of a primitive (and therefore of infinite primitives) is continuity in a closed and bounded interval [a,b].
From this it follows that the derivative of an integral of a function is the function itself, while the integral of the derivative is defined up to the arbitrary constant.
The following notable indefinite integrals can be defined :
The properties ofadditivity, monotonicity, and linearity of definite integrals also apply to definite ones.
For the search of the primitives, and therefore for the resolution of the integral calculus, one can go back to the notable formulas above, but also use two different methods given by integration by parts and by integration by substitution.
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The integration by parts takes its cue from the derivation rule of Leibnitz and from the theorem of integral calculus, in particular we have:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 22.04.2023
ISBN: 978-3-7554-3989-9
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