## Integral calculus

### Teaching unit

### Description

The integral calculus is closely related to the differential calculus, and is composed, a priori, of two fundamental problems: the calculation of primitives or anti-derivatives, and the calculation of areas or, more generally, volumes of regions. The well-known fundamental theorem of the calculation relates these two problems, therefore, the calculation of primitives is also usually presented as an indefinite integral and the calculation of areas as a definite integral.

The integral calculation not only applies to the calculation of areas, but in many other contexts, such as for calculating the average value of a function, or as a tool to solve certain types of differential equations. Therefore, together with the calculation of derivatives, it constitutes a basic tool to define and solve some types of differential equations, such as those of separable variables. On the other hand, the differential equations will be used to approach the study of some continuous models of interest in environmental sciences.

### Objectives

Possessing the basic mathematical knowledge that is necessary for theunderstanding of problems applied to experimental sciences and, especially, to environmental sciences.

Recognizing and distinguishing between discrete models and continuous models.

Formulate mathematically some applied problems, analyzing and correctly interpreting the results obtained.

Recognizing and using the derivative and the integral for the treatment and resolution of basic continuous models.

Using mathematical software in solving applied problems in environmental sciences.

### Subjects

#### Lecture topics

- The indefinite integral. Integration techniques.
- Calculation of integrals.
- The definite integral. Applications.
- Continuous models. Introduction to differential equations.

#### Laboratory topics

- Integrals and differential equations. Applications. (Computer lab)
- Exercises and problems. (Classroom)