Integral Calculus

Grading of the course requires grading of the following courses:

Linear Algebra and Differential Calculus (17BKLAD)

Lecturer:

Eva Feuerstein (gar.)

Tutor:

Eva Feuerstein (gar.)

Supervisor:

Department of Natural Sciences

Synopsis:

Definite and indefinite integral, methods of solutions, applications of definite integral for area/volume under curve, volumes and areas of rotational bodies, static moments and centers of gravity. Differential and difference equations and methods of their solution. Integral transformation, Laplace transformation.

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.

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

12. Z-transform - definition, properties and examples.

13. Inverse Z-transform, Test No. 2

14. Z-transform for solving linear difference equations.

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]Frank Ayres, Elliott Mendelson: Theory and Problems of Differential and Integral Calculus, Schaums Outline Series, 1990

[2]Edmund Landau: Differential and Integral Calculus, AMS Chelsea Publishing, 2001

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

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