Integral Calculus

Grading of the course requires grading of the following courses:

Linear Algebra and Differential Calculus (17PBBLAD)

Lecturer:

Tutor:

Supervisor:

Department of Natural Sciences

Synopsis:

The subject is an introduction to integral calculus and integral transforms.

Integral calculus: definition of an indefinite integral, properties and methods of integration (integration by substitution, integration by parts, partial fractions), definite integral, properties, Newton-Leibnitz fundamental theorem, simple applications of both indefinite and definite integrals, improper integral, solving differential equations (ODEs) (1st order ODEs with separable variables, linear 1st order homogenous as well as non-homogenous ODE?s, 2nd order linear ODEs homogenous and non-homogenous with constant coefficients),intro to multiple integrals, particularly applications of double integral.

Integral transform: Laplace transform - definition and properties, inverse Laplace transform, application of Laplace transform for solving nth order linear ODEs with constant coefficients,

Z-transform - definition and properties, inverse Z-transform, application of Z-transform for solving nth order linear difference equations.

Requirements:

Assessment:

Maximum 3 absences during the semester for serious reasons like sickness, injury etc. (medical certificate required).

Minimum 50% (i.e. 10 pts) evaluation at each of the 2 tests, each test consisting of 4 tasks, a task evaluated max. 5 pts each (the tests are taken in 6th and 13th week of the semester).

Exam:

1. Assessment recorded in „KOS? and signed by respective teacher in student?s “Index?,

2. Minimum 50% evaluation at the exam test. Exam test comprises of 10 tasks, a single task evaluated max 10% each.

Evaluation scale: less than 50% - F, 50-59% - E, 60-69% - D, 70-79% - C, 80-89% - B, 90-100% - A.

Syllabus of lectures:

1. Introduction to indefinite integral, basic properties, elementary functions integration, integration by parts, integration by substitution.

2. Rational functions integration, partial fraction technique.

3. Integration of trigonometric functions, combined techniques of integration.

4. Introduction to definite integral, simple geometrical applications (area, volume of rotational bodies, curve length).

5. Improper integral, introduction to differential equations, general solution.

6. Differential equations, initial value problem for ODEs, 1st order ODE with separable variables, linear 1st order ODEs homogenous and non-homogenous, method of variation of constant, homogenous ODEs (substitution z=y/x). Test No. 1.

7. nth order linear ODEs with constant coefficients and their solution.

8. Double integral, introduction and elementary methods of its calculating.

9. Jacobian and substitution in double integral, polar coordinates, geometrical applications of double integral.

10. Laplace transform- definition, properties and examples.

Syllabus of tutorials:

1. Elementary functions integration, integration by parts, integration by substitution.

2. Rational functions integration, partial fraction technique.

3. Integration of trigonometric functions, combined techniques of integration.

4. Definite integral, simple geometrical applications (area, volume of rotational bodies, curve length).

5. Improper integral, simple examples of improper integrals due to the function or due to the infinite interval of integration, introduction to differential equations, general solution.

6. 1st order ODE with separable variables examples, linear 1st order ODEs homogenous and non-homogenous, method of variation of constant, examples.

7. Homogenous ODEs (substitution z=y/x), nth order linear ODEs with constant coefficients and their solution.

8. Double integral, introduction and elementary methods of its calculating.

9. Jacobian and substitution in double integral, polar coordinates, geometrical applications of double integral.

10. Laplace transform properties and examples.

11. Inverse Laplace transform and application of Laplace transform for solving IVP for homogenous and non-homogenous nth order linear ODEs with constant coefficients.

12. Z-transform properties and examples.

13. Inverse Z-transform. Simple examples.

14. Z-transform for solving linear difference equations.

Study Objective:

The goal of the subject is to gain necessary theoretical background in the field of integral calculus and introduction to integral transforms. By the end of the course students will be able to apply the knowledge in solving various practical examples and problems of fundamental integral calculus, and integral transforms.

Study materials:

[1] Tkadlec J.: Diferenciální a integrální počet funkcí jedné proměnné, skriptum ČVUT, 2004

[2] Tkadlec J.: Diferenciální rovnice, Laplaceova transformace, skriptum ČVUT, 2005

[3] Hamhalter J., Tyšer J.: Integrální počet funkcí více proměnných, skriptum ČVUT, 2005

[4] http://math.feld.cvut.cz/mt/index.htm

[5] http://math.fme.vutbr.cz

[6] http://www.studopory.vsb.cz

[7] http://dagles.klenot.cz/rihova

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bakalářský studijní obor Biomedicínský technik - prezenční (compulsory course)