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Maths Coursework on Integration

Extracts from this document...

Introduction

Maths Coursework

Falk Konstantin Luebbe

Shady Areas

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 functionimage00.png. (Graph below)

image01.png

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 formulaimage31.png.

In order to find h the length of the area, which is 1, is dived by the number of trapeziums used. Hence,

image40.png

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 (image00.png).

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.

image59.png

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:

image67.png

...read more.

Middle

image45.pngimage46.png

image47.pngimage48.png

image49.pngimage50.png

image51.pngimage52.png

image53.pngimage54.png

image55.pngimage56.png

image57.pngimage58.png

Now, we put those values into the general formula in order to find the approximate area under the curveimage06.png:

image41.png

Inserting the values gives us the following:

image60.png

To solve the value in the bracket, we’ll use the GDC giving

A=0.25(7.92)

A=1.98 image28.png

We’ll continue with the function image08.png

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.

image61.png

image62.png

image63.png

image64.png

image65.png

image66.png

image68.png

image69.png

image70.png

Now, we put those values into the general formula in order to find the approximate area under the curveimage08.png :

image41.pngimage71.png

Inserting the values gives us the following:

image72.png To solve the value in the bracket, we’ll use the GDC

A=0.25(32.98)

A=8.245 image28.png

We will continue with the function image10.png

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.

image73.png

image75.png

image76.png

image77.png

image78.png

image79.png

image80.png

image81.png

image82.png

Now, we put those values into the general formula in order to find the approximate area under the curve image10.png :

image41.pngimage71.png

Inserting the values gives us the following:

image84.png

To solve the value in the bracket, we’ll use the GDC

A= 0.25(18.75)

A= 4.69 image28.png

...read more.

Conclusion

image24.png

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.

...read more.

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