Course syllabus for Integral calculus and ordinary differential equations

• define and compute upper and lower Riemann sums.
  
• define the concept of the integral and explain the connection between the derivative and the integral.
• define and compute generalized integrals.
• apply and motivate methods for computing integrals, both analytically and numerically.
•  explain the meaning of an ordinary differential eqation (ODE) and determine the order of the equation and whether its linear or non-linear.
• determine if an ODE of first order has unique solution. 
  
• apply and motivate analytical methods for solving ordinary differential equations.
• derive Laplace transforms and use the transform to solve ODEs. 
• define matrix and vector and their algebraic properties.
• reformulate higher order ODEs as a system of first order ODEs.
• define elementary functions as solutions to ODEs.
  
• implement numerical methods to solve ODEs and determine which method is appropriate for which problem.
  
• determine convergence order and stability region for numerical methods.
  
• combine knowledge of different concepts in practical problem solving.

The two main themes of the course are the integral and ordinary differential equations. Equal emphasis is put on the three pillars mathematical theory, analytical techniques and numerical methods.

Integral: Antiderivatives and integrals, methods of integration, integration of rational functions, generalized integrals, numerical integration. Applications of the integral: area, volume, center of gravity, arc length, area and volume of solids of rotation.

Ordinary differential equations:  general non-linear frirst order system, existence and uniqueness of solution, analytical techniques for separable and linear ODEs, second order linear ODEs with constant coefficients, Laplace transform, matrices and vectors algebraic properties, system of ODEs and higher order ODEs as systems of first order, numerical methods for ODE, convergence and stability, the finite element method.