Shady Areas
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Introduction
Maths Internal Assessment: Shady Areas Introduction: In this investigation you will attempt to find a rule to approximate the area under a curve (i.e. between the curve and the x - axis) using trapeziums (trapezoids). Let us consider using the function: . Before finding the area under the curve using trapeziums. I will use integration to help me find the exact area. This will enable me to compare the results I acquire using the trapezium method and to find out whether the use of more trapeziums will provide me with a more accurate estimation of the area under the graph. Using the integration method, I have found the exact area for the function is I will now use the trapezium method to find the approximation of the area under the curve. I have used Autograph to show the graph for the function of g. It can clearly be seen from the graph that the area under the curve is roughly by the sum of the areas of two trapeziums. ...read more.
Middle
of the second trapezium, I will replace 'a' and 'b' with the '' and '' values that I had calculated earlier but I will show my working: A= A= A2= 1.8125 As you know that A1 + A2 is equal to the area under the curve. So the area under the curve is equal to 1.5625 + 1.8125 = 3.375 The difference between the exact value previously done using the integration method and the estimated area under the curve using the trapeziums = . This shows us that the estimates using trapeziums is an over-estimate of the actual area (the estimated area under the curve of the two trapeziums is greater than the actual area under the curve). I am now going to increase the number of trapeziums to five and find second set of estimates for the area under the curve. This will tell me whether increasing the number of trapeziums will make the estimates more accurate. ...read more.
Conclusion
Area 0.652 Area for Fourth Trapezium 3.36 3.64 0.2 Area 0.7 Area for Fifth Trapezium 3.64 4 0.2 Area 0.764 Total Area 3.34 If the areas of the five trapeziums are added together, they give me the estimated area under the curve which is 3.34 0.604 + 0.62 + 0.652 + 0.7 + 0.764 = 3.34 The area given using the five trapeziums is still larger than the actual area found using the integration method ). Therefore it is an over - estimate of the actual area. However, the area obtained using five trapeziums are closer to the actual area than the area obtained using two trapeziums. The difference is not much. It can be said that using more trapeziums provides us with a lesser under/over estimate of the estimated area under the curve. To test this hypothesis, I will use 25,50,75 and 100 trapeziums to reach an estimate of the area under the curve. Using Autograph and Microsoft Excel 2007, I have created the following tables and graphs: ?? ?? ?? ?? Ritvik Menon H3CS ...read more.
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