Integral Calculus Project: Topics, Theory and Problem Solving

First Order Linear Differential Equations

The purpose of this project is to provide an opportunity to explore a topic in integral calculus or one of its applications in engineering science. Besides your project report, the final submission for the project also involves solving a set of three problems related to the topic you have explored in the body of your project. These problems, which are from your course textbook, are assigned to you by your professor based on your program at the time the project topic is assigned.

You may build your report based on the assumption that what you have learned in the earlier math courses you took at Sheridan need not be elaborated in your report. Any concepts other than what you have learned earlier in your Sheridan math courses, need to be elaborated in your report.

Special attention is paid to the consistency of the derivation of formulae and concepts in your report. Once the mathematical foundations are laid in a proper way, you need to introduce the topic you have been assigned from your program where the mathematical concepts you have explored are applied. It is important you support your derivations and conclusions by real world engineering applications.

You will apply the mathematical topic developed in your project to a given set of problems which you have been assigned by your instructor.

The report should consist of:

• A cover page including Student Name, Title, Date and Abstract.
  
  
• Thorough discussion of the mathematical topic, including the underlying concepts and derivations.

• Introduction to the engineering application relevant to your program, assigned by your instructor. Examples of the application of the concepts should be included.

• Solutions of the problems assigned by your professor, showing appropriate details of calculations.

• Bibliography of all cited sources.

The evaluation is based on the complexity and thoroughness of the study, as well as the rigor of method and accuracy of results. The project must be typed using word processing technology and should be submitted in PDF format to the appropriate dropbox on SLATE, as advised by your professor.

First order linear differential equations in general form, equations reducible to linear form (without using Laplace transforms)

Applications:

1. Series RL circuits [electronics]

1. Series RC circuits [electronics]

2. Second Order Linear Homogeneous and Non-homogeneous Differential Equations with Constant Coefficients with Applications

Second order linear homogeneous equations in general form, discussion of all possibilities for the roots of the auxiliary equation, the nonhomogeneous case (without using Laplace transforms)

Applications:

1. RLC circuits [electronics]
2. Damped motion of an object in a fluid [mechanical, electromechanical]
3. Forced Oscillations [mechanical, electromechanical]

3. Numerical Solution of First Order Linear Differential Equations with Applications Modified Euler’s Method, Runge Method

Applications:

1. Solving linear first order equations numerically. (Students may write a code to accompany their project.) [mechanical, electromechanical, electronics]

4. Numerical Solution of Second Order Linear Differential Equations with Applications Modified Euler’s Method, Runge-Kutta Method

Applications:

1. Solving linear second order equations numerically. (Students may write a code to accompany their project.) [mechanical, electromechanical, electronics]

5. Laplace Equation in Rectangular, Polar and Spherical Coordinates in One dimension with Applications

Laplace Equation and its meaning, the form of Laplace equation in different coordinate systems, symmetry and reduction to one dimension

Applications:

1. Heat transfer through a wall of infinite area, heat transfer through a cylindrical pipe of infinite length, heat transfer through a sphere [mechanical, electromechanical]
2. Mass transfer [mechanical, electromechanical]

Definition of moment of inertia, moment of inertia for a strip about center of mass, moment of inertia of an arbitrary plane area about the x-axis, y-axis, moment of inertia of a volume of revolution about the x-axis, y-axis.

Applications:

1. a) Calculating moment of inertia [mechanical, electromechanical]

Definition of centroid, centroid of a strip, centroid of an arbitrary plane area about the x-axis, y-axis, centroid of a volume of revolution about the x-axis, y-axis.

Applications:

1. a) Calculating centroid [mechanical, electromechanical]