Exercises of Double and Triple Integrals E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
double integrals
triple integrals
Initial theoretical hints are also presented to make the performance of the exercises understandable

Leseprobe

Table of Contents

“Exercises of Double and Triple Integrals”

INTRODUCTION

DOUBLE INTEGRALS

TRIPLE INTEGRALS

“Exercises of Double and Triple Integrals”

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

double integrals

triple integrals

Initial theoretical hints are also presented to make the performance of the exercises understandable

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 – DOUBLE INTEGRALS

Exercise 1

Exercise2

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

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II – TRIPLE INTEGRALS

Exercise 1 _

Exercises or2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

INTRODUCTION

In this exercise book, some examples of calculations relating to double and triple integrals are carried out.

These integrals represent the most used operations for real functions with several variables especially in physics and technology.

In order to understand in more detail what is explained in the resolution of the exercises, the reference theoretical context is recalled at the beginning of each chapter.

What is exposed in this workbook is generally addressed in advanced mathematical analysis courses (analysis 2) and, as such, a knowledge of at least the main properties of real functions with several variables is required, such as the concepts of mixed derivatives, differentiability and the Jacobian formalism .

I

DOUBLE INTEGRALS

Given a function of several real variables, it is possible to define the multiple integral according to Riemann , ie the integral carried out in each of its variables.

The theorems of the integral mean and the weighted mean as well as the properties of monotonicity,additivity and linearity always hold.

The notation is the same except to make explicit the writing for double and triple integrals:

Similarly, improper integrals maintain their properties and the methods of solving a multiple integral are the same, almost always accompanied by attempts to reduce the multiple integral into n different and successive integrations with respect to the n single variables.

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The reduction formulas are based on the assumption of a corollary of Fubini's theorem according to which if the integrand function, for example of two real variables, can be expressed as the product of two functions defined on a single variable, then also the integral double can be reduced to the product of two simple integrals:

Where the set T is given by the Cartesian product of the sets A and B.

This corollary guarantees this result for functions where the integral of the absolute value converges; Tonelli's theorem guarantees the same result for positive functions.

Such a reduction is also possible if the integrand function is continuous and the integration set is bounded.

If a change of variable is made in the calculation of multiple integrals, the following relation holds:

The determinant of the Jacobian matrix relating to the function after the change of variable appears in the formula; in polar and cylindrical coordinates this determinant is equal to r, in three-dimensional spherical coordinates it is equal to .

Exercise 1

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Calculate the following double integral over the specified set: