Integral Calculus
Code | Completion | Credits | Range | Language |
---|---|---|---|---|
F7ABBITP | Z,ZK | 4 | 2P+2C | English |
- Relations:
- In order to register for the course F7ABBITP, the student must have successfully completed or received credit for and not exhausted all examination dates for the course F7ABBLAD. The course F7ABBITP can be graded only after the course F7ABBLAD has been successfully completed.
- The course F7ABBFVP can be graded only after the course F7ABBITP has been successfully completed.
- Course guarantor:
- Tomáš Parkman
- Lecturer:
- Jiří Neustupa
- Tutor:
- Lukáš Liebzeit, Tomáš Parkman
- Supervisor:
- Department of Natural Sciences
- Synopsis:
-
The subject is an introduction to integral calculus and integral transforms.
Integral calculus: anti-derivative, indefinite integral, properties and methods of integration (integration by parts and by substitution, 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 ODEs, 2nd order linear homogenous and non-homogenous 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:
-
Form of verification of learning outcomes and other requirements for the student:
A - Compulsory attendance at exercises, absences must be duly excused in advance and subsequently documented e.g. by a medical certificate. Maximum of three properly excused absences. Active participation in the exercises is evaluated (maximum 5 points per semester, and these are added to the exam grade).
Attendance at lectures is not compulsory, but if a student fails to attend a lecture, he/she is obliged to supplement the material by self-study and must come prepared to the exercise.
B - Knowledge in the scope of individual topics of lectures is tested by two semester tests, which students take together in a single term in the middle and at the end of the semester according to the teaching schedule of the subject for the given academic year. Calculators certified for the mathematics graduation exam (i.e., non-programmable, without integrals and equation solving) and a list of formulas that will be included in the test assignment are allowed in the tests.
A condition for the award of credit is the fulfilment of point A and the achievement of at least 50% of the points from both semester tests (each test has a maximum of 40 points, the minimum for successful completion is 40 points in total).
A condition for admission to the examination is the credit entered in KOS. The exam is written only, lasts 120 minutes, calculators and formula lists are allowed, the same as for the mid-semester tests. The exam contains mostly numerical examples supplemented by theoretical subquestions within the of the material covered in lectures. The examination test consists of numerical problems from the material covered in the lectures and exercises supplemented by theoretical sub-questions. The maximum score is 75 points, for successful completion of the examination test, the student must obtain at least half of the points (i.e. 37.5 points). The points from both semester tests will be added to the exam score as follows: points above the mandatory 50% divided by 2 (max. 20 points) and points for activity in the exercises (max. 5 points). The total number of points is therefore (75+20+5) 100
Course grade: A: 100-90, B: 89-80, C: 79-70, D: 69-60, E: 59-50, F: less than 50.
- Syllabus of lectures:
-
1. Antiderivative, indefinite integral, basic properties. Table integrals, integration by parts.
2. Method of substitution. Integration of simpler rational functions - decomposition to partial fractions.
3. Integrals of partial fractions. Integration of trigonometric functions, their products and powers.
4. Definite (Riemann) integral. Geometric and physical meaning. Newton-Leibniz formula. Geometric and physical applications.
5. Improper Riemann integral (of unbounded functions or/and on unbounded intervals).
6. Functions of two variables. Graph, limit, continuity, partial derivative. Tangent plane, differential. Directional derivative.
7. 1st mid-term test.
8. Double integral. Methods of calculation: Fubini's theorem, transformation to polar coordinates. Geometric and physical applications.
9. First order ordinary differential equations (ODEs) with separable variables. Initial condition, Cauchy problem. Simple applications.
10. Linear first order ODEs (homogeneous and inhomogeneous). Variation of constant method.
11. Linear 2nd order ODEs with constant coefficients (homogeneous and inhomogeneous). Characteristic equation, fundamental system of solutions of the homogeneous equation, general solution of the homogeneous equation.
12. Particular solution of an inhomogeneous equation, general solution of the inhomogeneous equation. How to find a particular solution: method of estimation and variation of constants.
13. Laplace transform and inverse Laplace transform. Application of Laplace transform in solving ODEs.
14. Repetition: solving different types of ODEs by different methods. Applications in geometry, physics, chemistry, population dynamics, spread of a contagion, spread of an information.
- Syllabus of tutorials:
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1. Indefinite integrals, application of formulas from the table of basic indefinite integrals, integration by parts.
2. Method of substitution for indefinite integrals. Decomposition of simple rational functions to partial fractions.
3. Integration of simpler rational functions. Integration of trigonometric functions, their products and powers.
4. Calculation of the definite integral, Newton-Leibniz formula, integration by parts and method of substitution in definite integral.
5. Geometric and physical applications of definite integral: area of a plane plate, length of a curve, volume and surface area of a body of revolution, mass, coordinates of the center of gravity, etc.
6. Calculation of improper integrals.
7. Functions of two variables: calculation of partial derivatives, tangent plane, directional derivatives.
8. Calculating the double integral: application of the Fubini theorem and transformation to polar coordinates. Geometric and physical applications of the double integral.
9. First order ordinary differential equations (ODEs), solution by separation of variables.
10. Linear 1st order ODEs, method of variation of a constant.
11. Linear 2nd order ODEs with constant coefficients. Characteristic equation, fundamental system of solutions of the homogeneous equation, general solution of the homogeneous equation.
12. General solution of the inhomogeneous equation, methods of finding particular solutions: variation of constants and method of estimation. Physical applications (in mechanics, electrodynamics, etc.).
13. Application of Laplace transform in solving ODEs.
14. Repetition and systematization of knowledge before the exam.
- Study Objective:
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The goal of the course is to gain basic knowledge in the fields of integral calculus (indefinite integral, definite Riemann's integral, double integral, applications) and ordinary differential equations (= ODEs) (ODEs with separable variables, 1st and 2nd order linear ODEs with constant coefficients, applications).
- Study materials:
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Study materials
[1] Neustupa J.: Mathematics 1, skriptum ČVUT, 2004
[2] Bubeník F.: Problems to Mathematics for Engineers, skriptum ČVUT, 2007
[3] Stewart, J. : Calculus, Brooks/Cole, 2012
[4]
- Note:
- Time-table for winter semester 2024/2025:
- Time-table is not available yet
- Time-table for summer semester 2024/2025:
-
06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon Tue Wed Thu Fri - The course is a part of the following study plans:
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- Prospectus - bakalářský (!)
- Biomedical Technology (compulsory course)
- Biomedical Technology (compulsory course)