ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Center Faculty of Mathematics
Call: Second Semester
Teaching: Sin Docencia (En Extinción)
Enrolment: No Matriculable (Sólo Planes en Extinción)
- To introduce students with the essential support of examples and practice in the construction and understanding of the concept of Riemann integral of real bounded functions on compact intervals.
- To know and prove the main properties of the Riemann integral, and to check if a given function is integrable or not.
- To understand the relationship between the differential and the integral calculus, established by the Fundamental Theorem of Calculus. To obtain primitives and to calculate integrals by application of the rule of Barrow .
- To apply integral calculus in order to solve different geometrical problems.
- To use a package of symbolic calculation with application to the integral calculus.
1. The Concept of Riemann Integral of a limited Function in a Compact Interval: Equivalence Formulations. Examples of Riemann-Integral Functions
Partitions of a Compact Interval.
Riemann Sums.
Concept of Riemann Integral of a Limit Function in a Compact Interval.
Intuitive Interpretation of the Integral.
Higher Sums and Lower Sums.
Higher Integral and Lower Integral.
Equivalence Formulations of the Integral Function Concept.
Examples of Integral Functions: Integrability of the Continuous Functions and the Monotonous Functions.
2. Properties of the Integral and Integral Functions
Linearity of the Integral.
Additive Effect of the Integral in relation to the Interval of Integration.
Monotone of the Integral. Modular Bound.
Averages. The Theorem of the Mean Value of the Integral Calculus.
3. The Fundamental Theorem of the Calculus
Concept of Primitive.
First Formulation of the Fundamental Theorem (generalization of the Barrow Rule).
The “Integral Function” of a Riemann Integral Function.
Second Formulation of the Fundamental Theorem.
Theorems of Change of Variable and Integration by Parts for a Riemann Integral.
4. Indefinite Integral
Concept and Properties.
Primitive Calculus by Parts and by Change of Variable.
Methods of Calculus of Elemental Primitives.
5. Applications of the Riemann Integral
Areas Calculus of some plane figures.
Volumes Calculus of Solids of Revolution.
Length Calculus of Graphics of Regular Functions.
Lateral Areas Calculus of Revolution Fields.
Basic Bibliography
ABBOTT, S. (2015) Understanding Analysis. Springer (SpringerLink eBook Collection – Mathematics & Statistics, https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-1-4939-2712-8)
APOSTOL, T. M. (1977) Análisis Matemático. Reverté.
BARTLE, R. G., SHERBERT, D. R. (1999) Introducción al Análisis Matemático de una variable (2ª Ed.). Limusa Wiley.
Complementary Bibliography
LARSON, R., HOSTETLER, R. P., EDWARDS, B. H.: Cálculo (8ª Ed.). McGraw-Hill. 2006
MAGNUS, R.: Fundamental Mathematical Analysis. Springer. 2020. (SpringerLink eBook Collection – Mathematics & Statistics, https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-3-030-46321-2)
PISKUNOV, N.: Cálculo Diferencial e Integral. Montaner y Simón. 1978
SPIVAK, M.: Calculus. Reverté. 1978
In addition to contribute to achieve the general and transverse competences taken up in the memory of the degree, this subject will allow the student to get the following specific competences:
CE1 - To understand and use mathematical language;
CE2 - To know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To Identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstrac properties and material facts of a problem, distinguishing them from those purely occasional or incidental;
CE9 - To use statistical analysis applications, numerical and symbolic computation, graphical visualization, optimization and scientific software, to experience and solve problems in mathematics.
Non-teaching course
The virtual course or the Teams platform will be used as the mechanism to provide students with the necessary resources for the development of the subject (explanatory videos, notes, exercise bulletins, etc.).
The tutorials will be face-to-face
The final exam will measure the knowledge obtained by students in relation to the concepts and results of the subject, both from a theoretical and practical point of view, also assessing clarity, the logical rigor demonstrated in their exposition. The achievement of the basic, general and specific competences to which allusion is made in the Memory of the Degree in Mathematics of the USC and which were previously indicated will be evaluated.
Warning. In cases of fraudulent performance of tests or plagiarism (plagiarism or improper use of technologies), the provisions of the regulations for evaluating the academic performance of students and reviewing grades will apply.
TOTAL HOURS (150)
150 hours: 0 hours of presence work in the class and 150 hours of personal work of the student.
PRESENCE WORK IN THE CLASS (NA)
PERSONAL WORK OF THE STUDENT (150 hours)
Personal work will depend on the students. On average, 150 hours per student are estimated.
To have studied the subject "Introduction to the Mathematical Analysis" and to attend or have attended the subject "Continuity and Derivability of One Real Variable Functions".
Francisco Javier Fernandez Fernandez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813231
- fjavier.fernandez [at] usc.es
- Category
- Professor: University Lecturer
| 06.03.2026 10:00-14:00 | Grupo de examen | Classroom 06 |
| 07.10.2026 10:00-14:00 | Grupo de examen | Classroom 06 |