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Maths Coursework on Integration

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Introduction

Maths Coursework

Falk Konstantin Luebbe

SL Type 1

Introduction:

In this coursework we are trying to find a way to approximate the area under a curve using trapeziums. In order to do this we are searching for a formula to express the approximation of the area using n numbers of trapeziums

Consider the function . (Graph below) The diagram above shows the graph of g. In order to find an approximate area under the curve between the x values 0 and 1 we are going to use the trapezium rule. We are going to start with two trapeziums, add them together to find the approximate area under the curve.

To find the area of a trapezium we will use the formula .

In order to find h the length of the area, which is 1, is dived by the number of trapeziums used. Hence, In order to find the heights a and c for the first- and a and c for the second trapezium assuming that c of the first and a of the second are equal, we’ll use the equation of the graph ( ).

The three different heights, or y-values, can be expressed by

g(0), g(0.5), g(1)

Now, in order to find the corresponding y-values, we’ll insert the three x-values into the equation. Now we’ve got the corresponding heights and are able to calculate the approximate area under the curve

The formula for the area under the curve in this case would be: Middle              Now, we put those values into the general formula in order to find the approximate area under the curve : Inserting the values gives us the following: To solve the value in the bracket, we’ll use the GDC giving

A=0.25(7.92)

A=1.98 We’ll continue with the function First we have to calculate the y-values in order to get the heights of the trapeziums by inserting the x-values into the formula.         Now, we put those values into the general formula in order to find the approximate area under the curve :  Inserting the values gives us the following: To solve the value in the bracket, we’ll use the GDC

A=0.25(32.98)

A=8.245 We will continue with the function First we have to calculate the y-values in order to get the heights of the trapeziums by inserting the x-values into the formula.         Now, we put those values into the general formula in order to find the approximate area under the curve :  Inserting the values gives us the following: To solve the value in the bracket, we’ll use the GDC

A= 0.25(18.75)

A= 4.69 Conclusion If we want to calculate the area under the curve from for example x=-1 to x=1 the result is invalid. This is because the two parts of the curve never touch x=0; they get closer and closer but never touch it. Therefore it’s not possible to calculate the area under the curve of y=x^-1.

Concluding we can say that the more turning points are included in our calculation of the area under a curve the more trapeziums are needed to give a decent approximation. That means that our general formula can give a very inaccurate, even a completely false result if it’s used in the wrong way and too few trapeziums are used. Therefore, the formula always has to be adapted at the shape of the graph; if the graph is a straight line, it’s sufficient to use only one trapezium. But if the area under the curve includes many stationary values, many trapeziums are necessary to produce an accurate approximation as seen in the last example.

In the end we can state that the accuracy of our general formula depends on the shape of the graph. However even quite complex curves with more than one turning point can be solved accurately using enough trapeziums; the more trapeziums we use the more accurate result we get. But nevertheless, it takes way too much time to calculate 100 or even more trapeziums with our formula. Therefore we can say that our formula is not suitable for the calculation of the area of curves with more than two stationary values.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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