Course syllabus adopted 2026-02-23 by Head of Programme (or corresponding).
Overview
- Swedish nameIntegralkalkyl
- CodeMVE780
- Credits7.5 Credits
- OwnerTKIEK
- Education cycleFirst-cycle
- Main field of studyMathematics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language Swedish
- Application code 51141
- Open for exchange studentsNo
- Only students with the course round in the programme overview.
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0126 Project 1.5 c Grading: UG | 1.5 c | ||||||
| 0226 Laboratory 1.5 c Grading: UG | 1.5 c | ||||||
| 0326 Examination 4.5 c Grading: TH | 4.5 c |
In programmes
Examiner
- Mattias Lennartsson
- Part-time fixed-term teacher, Algebra and Geometry, Mathematical Sciences
Eligibility
General entry requirements for bachelor's level (first cycle)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Specific entry requirements
The same as for the programme that owns the courseApplicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements
Course specific prerequisites
Introductary course in mathematics.Aim
The purpose of the course is to, together with the other math courses in the program, provide a general knowledge in the mathematics required in further studies as well as in the future professional careerLearning outcomes (after completion of the course the student should be able to)
Explain and apply the concept of the Riemann integral for functions of one real variable, understand the interpretation of the integral as area, use the Fundamental Theorem of Calculus, and give an overview of improper integrals and associated notions of convergence. Use classical analytical methods and basic numerical methods to compute integrals, and use integrals to determine quantities such as arc length, area, volume, and mass-related quantities in simple models.
Formulate ordinary differential equations of first and higher order as mathematical models for time-dependent phenomena, give an overview of existence and uniqueness, and solve basic classes of ordinary differential equations using analytical methods.
Formulate and analyse systems of first-order ordinary differential equations in vector and matrix form, and reduce higher-order equations to such systems.
Formulate initial value problems for ordinary differential equations, and choose, apply, and give a basic analysis of fundamental numerical methods for their solution, with respect to, for example, order of convergence, stability, and preservation of relevant quantities.
Implement and use numerical algorithms in Python for computations within the scope of the course, and analyse, interpret, and communicate in writing modelling choices, computational results, and limitations in engineering applications.
Content
Integrals
Area as a limit; Riemann sums; definition of the Riemann integral; Mean
Value Theorem for integrals; Fundamental Theorem of Calculus; improper integrals,
absolute convergence, and the comparison test.
Computation of integrals
Change of variables (including inverse substitution); integration by parts; integration of rational functions via partial fractions; arc length, area, and volume; solids of revolution; numerical integration (midpoint, trapezoidal, Simpsons rule); error estimates and convergence.
Ordinary differential equations
Classification of differential equations (ordinary/partial, order); existence and uniqueness (Picards theorem); first-order ODEs (separable equations by separation of variables, linear equations with integrating factor); reduction from second-order to first-order ODEs; linear ODEs with constant coefficients; Eulers equation.
Systems of ordinary differential equations
Matrix and vector notation; systems of first-order equations with initial conditions; reduction of higher-order ODEs to systems of first-order equations; existence and uniqueness for systems; elementary functions defined via ODEs.
Numerical solution of ordinary differential equations
Initial value problems; explicit/implicit Euler methods and the midpoint method; order and stability analysis (stability region); energy-preserving properties.
Applications and programming
Python as a computational tool for integration and ODEs; implementation, testing, and validation; for example, computation of mass and centre of mass, and two- and three-body dynamics.
Organisation
Instruction is given in lectures and smaller classes. More detailed information will be given on the course web page before start of the course.Literature
S. Larsson, A. Logg, A. Målqvist, MATEMATISK ANALYS & LINJÄR ALGEBRA (II): Integralkalkyl och ordinära differentialekvationerExamination including compulsory elements
Written final exam at the end of the course.The laboratory component is examined through mandatory computer labs during the course.
The project component is examined through an intermediate test during the course.
Non-compulsory assignments may render bonus points to the exam. Information about this will occur on the course web page at the start of each course occasion.
The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers about disability study support.