Numerical Methods coursework

Numerical Methods Coursework

Numerical Integration

The Problem

Integration means finding the area underneath a particular region of a function. At my current knowledge of maths I am not able to integrate various functions. Therefore I am going to use knowledge of numerical methods to produce an approximation to an area which does not have an analytic solution.

In my coursework I will integrate the function:

The graph below shows the graph of the function.

The red arrow determines the region between 0.25 and 1.25, which then leads to the integral:

I can not solve this problem using the knowledge of C1 and C2, because I am not able to integrate cos(x) yet. Due to this I suggest that this problem will be appropriate for numerical solution.

The Approximation Rules

To solve this problem, I am going to use knowledge of numerical integration studied in the “Numerical Methods” textbook. The approximate methods of definite integrals may be determined by numerical integration using:

1. The Trapezium Rule:

The Trapezium Rule divides the area underneath the curve into trapeziums. We can then use the formula  (where a, b are the bases and h is the height of the trapezium) to estimate the area.

Dependent on the amount of trapezia (n) used the general formula is:

2. The Midpoint Rule:

The Midpoint Rule divides the area underneath the curve into rectangles. We can then use the formula (where a, b are the 2 different sides of the rectangle) to estimate the area.

Dependent on the amount of rectangles (n) used the general formula is:

3. Simpson’s Rule

Simpson’s Rule is a weighted average of  and . The normal average is exactly in the middle of  and . But in this rules the ...