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Elements of Mathtematics II

Code: PARC03     Sigla: EMII

Áreas Científicas
Classificação Área Científica
CNAEF Mathematics

Ocorrência: 2021/2022 - 2S

Ativa? Yes
Página Web:
Unidade Responsável: Departamento de Matemática
Curso/CE Responsável: Professional Technical Higher Education Courses in Networks and Information Systems

Ciclos de Estudo/Cursos

Sigla Nº de Estudantes Plano de Estudos Anos Curriculares Créditos UCN Créditos ECTS Horas de Contacto Horas Totais
TSPPAR 45 Plano de Estudos_2015_16 1 - 6 60 162

Docência - Responsabilidades

Docente Responsabilidade
Júlia Maria da Rocha Vilaverde Justino

Docência - Horas

Theorethical and Practical : 4,00
Type Docente Turmas Horas
Theorethical and Practical Totais 1 4,00
Júlia Maria da Rocha Vilaverde Justino 4,00

Língua de trabalho



The general objective of this course unit is to provide students with the basic mathematical knowledge required in the professional training of a top professional technician concerning integral calculus.

Resultados de aprendizagem e competências

By the end of term time, students should be able to:

  1. Identify and calculate antiderivatives of elementary functions.

  2. Calculate antiderivatives using integration by parts, integration by partial fraction decomposition and integration by substitution.

  3. Interpret the Riemann integral and its properties.

  4. Interpret the indefinitive integral and apply the fundamental theorems of calculus.

  5. Calculate areas and volumes of revolution by integration.

  6. Compute improper integrals.

Modo de trabalho


Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Previous mathematical knowledge acquired through the course unit Elementos de Matemática I, in particular differential calculus.


1. Antiderivatives 
1.1. Antiderivative: motivation, definition and properties. Antiderivatives of elementary functions.
1.2. Techniques of integration: by parts, by substitution and by partial fraction decomposition.

2. Integral Calculus 
2.1. Riemann integral and its geometric interpretation; properties.
2.2. Indefinitive integral and its properties. First and second fundamental theorems of calculus.
2.3. Integration by parts and by substitution.
2.4. Integral calculus' applications.
2.5. Improper integrals.

Bibliografia Obrigatória

Sérgio Fernandes; Cálculo Integral em ℝ, 2019

Bibliografia Complementar

Acilina Azenha e Maria Amélia Jerónimo; Elementos de Cálculo Diferencial e Integral em R e Rn, McGraw-Hill, 2000. ISBN: 9789728298036
Fernanda Sequeira; Análise Matemática 3º Volume - Exercícios Resolvidos e Propostos, Litexa Editora, 1982. ISBN: 9789725780497

Métodos de ensino e atividades de aprendizagem

Student-centered approach is the pedagogical methology applied. During classes, the fundamental concepts on the different subjects of the course unit are firstly presented, illustrated by some application examples. Afterwards, students will carry out exercises to consolidate knowledge on the covered topics through collaborative working group, under the guidance of the teacher.

Tipo de avaliação

Distributed evaluation with final exam

Componentes de Avaliação

Designation Peso (%)
Teste 100,00
Exame 0,00
Total: 100,00

Componentes de Ocupação

Designation Tempo (Horas)
Estudo autónomo 102,00
Frequência das aulas 60,00
Total: 162,00

Obtenção de frequência

The approval in this UC (curricular unit) can be obtained through two assessment processes: Continuous Evaluation or Exam Evaluation.

The Continuous Evaluation presupposes the accomplishment of 4 summative tests and a compulsory attendance of at least 75% of the classes.
Assigning by MT1, MT2, MT3 and MT4 the grades (from zero to 5 values, rounded to tenths) obtained in each of the 4 summative tests, the final classification CF (rounded to units) will be the plain sum of the four grades.

The approval conditions are as follows:

  1. If CF is greater than or equal to 10 and less than 18, the student passes on with a final grade equal to CF, provided that the classification in any of the sums P1=MT1+MT2 and P2=MT3+MT4 is greater than or equal to 3.5 values.

  2. If a student fails the approval conditions referred in point 1, the student can recover the lowest grade obtained in P1 and P2 by performing a recovery test on the date of the normal period exam, provided that P1 or P2 is greater than or equal to 3.5 values. 

Students who have not obtained approval for Continuous Assessment may take an exam, being approved as long as they obtain a grade of 10 or higher.

NOTE: In any of the evaluation processes, whenever the final classification is greater than or equal to 18 values, the student must carried out an oral test, obtaining as a final grade the average of the classifications of the written test and of the said oral test . If the student does not attend the oral test, the final classification will be 17 values.

Fórmula de cálculo da classificação final

CF = MT1 + MT2 + MT3 + MT4
Exam evaluation

Avaliação especial (TE, DA, ...)

Working students, high-level athletes, association leaders and students under the Religious Freedom Law must address, until the second academic week of the semester, to the head of the Curricular Unit to present their pertinent specificities, in accordance with the terms of the respective diplomas under penalty of failure to enforce them for lack of objective conditions.

Melhoria de classificação

In this course unit passed students of this academic year may only apply to their improvement classification in the supplementary period exam.


  • Each summative test shall be of 60 minutes, the recovery test of 90 minutes and each exam of 150 minutes.

  • To perform the recovery test and/or exams, an identification document with photo has to be presented.

  • During assessment tests and examinatiom, only the enquiry form given by the teacher is allowed; handling or displaying any electronic equipment is prohibited.

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