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The Area Under A Curve

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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.

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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


Trapezium 1 = 4+16 x 2 = 20


Trapezium 2 = 16 + 64 x 4 = 160


Trapezium 3 = 64 + 100 x 2 = 164


Triangle = 2x4 = 4


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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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.

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