Integral Calculus
Code  Completion  Credits  Range  Language 

17KBIITP  Z,ZK  5  12P+12C  Czech 
 Vztahy:
 The course 17KBIITP can be graded only after the course 17KBILAD has been successfully completed.
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Natural Sciences
 Synopsis:

The subject is an introduction to integral calculus and integral transforms.
Integral calculus: primitive function, indefinite integral, properties and methods of integration (integration by parts and by substitution, partial fractions), definite integral, properties, NewtonLeibnitz 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 nonhomogenous ODEs, 2nd order linear homogenous and nonhomogenous ODEs with constant coefficients),intro to multiple integrals, particularly double integral and applications.
Integral transforms: Laplace transform and inverse Laplace transform and their application for solving nth order linear ODEs with constant coefficients.
 Requirements:

Assessment:
Maximum 3 absences during the semester for serious reason.
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 7th and 13th week of the semester.
Exam:
1. Assessment recorded in KOS by respective teacher
2. Minimum 50pts evaluation at the exam test. Exam test comprises of
7 tasks, a single task evaluated max 10% each, together up to 70%
5 tests, evaluated max 2% each, together up to 10%
5 tests evaluated max 1% each, together up to 5%
plus points gained for assesment that are transferred for the exam evaluation min 5 max 15 pts.
Evaluation scale: A: 90100, B: 8089, C: 7079, D: 6069, E: 5059, F: less than 50.
 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 nonhomogenous, method of variation of constant, homogenous ODEs (substitution z=y/x).
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 nonhomogenous nth order linear ODEs with constant coefficients.
12. Ztransform  definition, properties and examples.
13. Inverse Ztransform, Test No. 2
14. Ztransform 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 nonhomogenous, 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 nonhomogenous nth order linear ODEs with constant coefficients.
12. Ztransform properties and examples.
13. Inverse Ztransform. Simple examples.
14. Ztransform 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 integral transforms and practical skills in solving various 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] Neustupa J., Kračmar, S.: Sbírka příkladů z Matematiky I., skriptum FS ČVUT
[5] Neustupa J.: Matematika I, skriptum FS ČVUT
[6] http://math.feld.cvut.cz/mt/index.htm
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Biomedical Informatics  combined study (compulsory course)