# Numerical integration coursework

by 06raymondm (student)

Numerical integration coursework

Problem

For this coursework, I am going to use my knowledge of numerical methods to produce an approximation to an area which does not have an analytic solution. I will be finding the approximation, to an appropriate degree of accuracy, of the integral shown above. On the graph below is the area that I will be approximating underneath the curve of y=

from x=0 to x=2. Note that throughout my method I worked in radians.

This problem is appropriate for numerical solution as I chose my graph to be a polynomial curve involving a square root so that there would be no analytical solution. Due to the fact that I cannot yet integrate functions like this approximating methods will have to be used. According to the numerical methods module the three approximation methods to be used are:

Mid-point rule- this method was adopted because it is used to approximate the area underneath the graph by dividing it up into individual rectangles.

Trapezium rule- this method was adopted because it is used to approximate the area underneath the graph, this is done so by dividing it up into individual trapeziums.

Simpson’s rule- I have realised that out of the two, the mid-point rule and the trapezium rule, the mid-point rule is more accurate, Simpson’s rule is therefore just an average that uses this fact by weighting the average towards the mid-point rule to give a more accurate answer.

Use of technology

I used Microsoft Excel throughout my method along with many algorithms, these algorithms allowed me to reach a more accurate approximation thanks to the fact that there were no rounding errors usually gained through using a calculator that gives answers to fewer decimal places. Overall then Excel is more accurate than the use of a calculator due to being able to work to more decimal places, it is also easier to use and saves a lot of time.

Mid-point Rule

The first step is to use the mid-point rule; this will be done by splitting the area under the curve into rectangular strips as shown by the diagram.

Thus far using the mid-point rule only gives a good estimate, however obviously as you increase the amount of strips the accuracy of the area increases. To estimate the area you first need to find the area of each rectangle. You can do this by thinking logically; to get the width of each rectangle you need to do the range between the limits of the integral divided by ...