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Integration

Paper, Order, or Assignment Requirements
 
 
Using a function which can’t be solved by integration. Using numerical theory to integrate function by using premezium and simpsons theory in exert produce graphs with data. My function that I have chosen is “sin(3×)/(×power2 + 4)power 2.
Part 3: Integration (15% of the module mark)
 
For this piece of coursework you must use the methods for numerical integration to produce approximations to an area which cannot be calculated exactly using other techniques covered in the module 4ET005.
Credit will be given for good knowledge of the methods themselves and the way that they behave in relation to error. By using a spreadsheet program you will be able to produce very accurate approximations to your chosen area and be able to judge how reliable your approximations are.
 
IMPORTANT:
Your chosen problem should not have an analytical solution and you need to demonstrate this using all the integration techniques you have covered in this module. You should explain why each technique does not work, rather than stating ‘this integral is really hard’.
 
This means that you shouldn’t choose an integral that can be solved using the techniques you have learnt in this module. For example, you will know how to evaluate the following integrals, so you shouldn’t pick anything similar for this coursework.
 
 
For example would an appropriate integral for this coursework, but is not.
 
What you need to do:
 
Phase 1 – Problem Specification
Make sure that you tell the reader exactly what you are attempting to do in your coursework. When you are trying to approximate an integral make sure that you tell the reader between which values you are integrating – show this on a diagram. You should briefly explain why your problem cannot be solved using normal analytical methods. See above. This phase should be fairly short.
 
Phase 2 – Strategy
Say what you are going to do to solve the problem. State which Numerical Methods you will use and the formulae involved in them. Where appropriate, you may wish to include a diagram to explain the formulae, but do not derive any of the standard formulae – you will earn no marks for deriving standard results.
 
Phase 3 – Formula Application
Do this on a spreadsheet. Try to set it out as clearly as you possibly can, so that the reader can see how all of the cells relate to one another. Remember to include the version of the spreadsheet where formulae are displayed somewhere in your report (perhaps in an appendix or maybe next to the spreadsheet itself). Label cells where appropriate. Try to stick to a standard notation.
 
Phase 4 – Use of technology
You will use Excel to devise your spreadsheets. You should say what its limitations are (e.g how many decimal places can it handle?). You should say how many decimal places your calculations are being made to.
 
Phase 5 – Error Analysis
This is a very important section. You should try to give an estimate of how much error there is in your solution to the problem. You may wish to analyse differences between successive approximations and then the ratio of differences, in order to see how fast your series of approximations in tending to the true value; you can use this to give an estimate of the error involved (see notes below). You could also use Maple to check the true error values.
 
Phase 6 – Interpretation
Relate the solution you have obtained back to the original problem. Maybe you will want to discuss how you could improve your estimate and how long this would take in terms the number of extra iterations required. If you think your results are as good as the calculating power of the spreadsheet will allow, you should say why you think this.


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