Integration of One Real Variable Functions

- 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 (9 hours CLE).
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 (5 hours CLE).
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 (5 hours CLE).
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 (4 hours CLE).
Concept and Properties.
Primitive Calculus by Parts and by Change of Variable.
Methods of Calculus of Elemental Primitives.

5. Applications of the Riemann Integral (5 hours CLE).
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.

The general methodological indications established in the USC Mathematics Degree Title Memory will be followed.

Teaching is programmed in theoretical classes, small group practices, small group computer practices and tutorials. In the theoretical classes the essential contents of the discipline will be presented, and will allow the work of the basic, general and transversal competences, in addition to the specific competences CE1, CE2, CE5 and CE6. In the interactive sessions, problems or exercises of more autonomous realization will be proposed, which will make it possible to emphasize the acquisition of specific competences CE3 and CE4. Finally, the tutorials will be devoted to discussion and debate with the students, and to the resolution of the proposed tasks with which it is intended that the students practice and strengthen knowledge. In the computer classes, the MAPLE program will be used as a study tool, thus working on the specific competence CE9.

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

In general, there will be an assessment that combines a continuous assessment with a final test.

The continuous assessment will be based on the results obtained in the intermediate exams made by the students, as well as the various activities carried out in this area. It will allow verifying the degree of achievement of the specific competences previously determined, with special emphasis on the transversal competences CT1, CT2, CT3 and CT5.

Regarding the final and second chance exam, it 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.

As previously mentioned, the evaluation will be carried out by combining a continuous evaluation with a final test.

The continuous assessment will consistof carrying out two intermediate tests. The mark of the continuous assessment (C), over 10 points, will be calculated using the following formula:

C=1/2*P1+1/2*P2,

with P1 and P2 the marks obtained in the intermediate tests.

Although the number and type of continuous assessment tests will be the same for the two theoretical groups, their content may vary depending on the theoretical group. The coordination and educational equivalence of the two exhibition groups is guaranteed.

With the mark of the continuous assessment (C), over 10 points, and the mark of the final face-to-face test (F), over 10 points, the final mark of the subject (NF) will be calculated using the following formula:

NF = max {F, 0.3 * C + 0.7 * F}

The final exam will be the same for both theoretical groups.

It will be understood as NOT PRESENTED who at the end of the teaching period is not in a position to pass the subject without taking the final exam and does not appear for said test.

In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be a test of the same type as the first.

The exam corresponding to the second opportunity will be the same for both theoretical groups.

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: 58 hours of presence work in the class and 92 hours of personal work of the student.

PRESENCE WORK IN THE CLASS (58 hours)

(CLE) Blackboard classes in big group (28 hours)

(CLIS) Interactive classes in reduced group (14 hours)

(CLIL) Interactive classes of laboratory/tutorials in reduced group (14 hours)

(TGMR) Small group tutorials or individualized (2 hours)

PERSONAL WORK OF THE STUDENT (92 hours)

Personal work will depend on the students. On average, 92 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".