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The Area Under A Curve

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Introduction

Juri Gregurec

Maths Coursework

The Area Under A Curve

My aim is to find the area under a curve on a graph that goes from –10 to 10 along the x axis and from 0 to 100 on the y axis. The curve will be the result of the line y=x . I will attempt several methods and improve on them to see which one gives the most accurate answer. The graph I am using looks like this: -

Counting Squares Method

The first method I will use to find the area is the counting squares method. For this method I will draw the graph on cm paper and estimate the amount of squares that the area under the curve takes up. To do this I will first count all the whole squares, and then count all the half squares and divide that number by two to give a rough estimate of the area under the curve.

Altogether I counted 10 whole squares and 14 half squares. When the half squares were divided by 2, the total number of squares was 17 squares.

...read more.

Middle

This method is still not entirely accurate but gives a more accurate value for the area under the curve than the counting squares method.

Counting Trapeziums

The method I have chosen to use now will increase the accuracy even more by splitting the curve up into trapeziums. I will split the curve up into 3 trapeziums and one triangle to find an even more accurate value for the area under the curve.

The formula I shall use to find the area of a trapezium is:

 Area   =   sum of parallel sides  x height

                      2

Trapezium 1 = 4+16 x 2 = 20

                   2

Trapezium 2 = 16 + 64 x 4 = 160

                       2

Trapezium 3 = 64 + 100 x 2 = 164

                      2

Triangle = 2x4 = 4

                2

Total = 20 + 160 + 164 + 4 = 348

Again I will multiply this by 2 to give the area for both sides of the parabola

348 x 2 = 696

This method is again, more accurate than the last but is still not precise because there is a lot of space that is left between the line of the trapezium, and the line of the curve. This is a problem that would make the area value to big.

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Conclusion

Conclusion

The first method I used was good for rough estimating but was very inaccurate when trying to calculate the area exactly. My second method although seemed again, a rough estimate, came closest to the result I achieved from Simpson’s rule. This may have been because although there was a lot of excess space, there was a gap at the top of the final rectangle, which may have made up for it. The next method I used was very inaccurate and gave a value that was too large. My next trapezium rule was all right and gave a closer value to the answer but the overall best method was Simpson’s Rule. It gave a value that was accurate because of using more trapeziums and therefore had the least excess space between the trapezium line and the curve. This is probably the closest result I could get when using straight-line formulae such as trapezium rules to calculate the area under a curve.

...read more.

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