- Level: International Baccalaureate
- Subject: Maths
- Word count: 1404
type 1 maths portfolio trapezium rule
Extracts from this document...
Introduction
There are different ways through which can estimating the area under a curve of a certain function and one of the many methods is by the trapezium rule.
The area of a trapezium (trapezoid) is determined by;
Or in words, the average of twosides times the base, which could also be expressed as where by if considering trapeziums formed under a curve; parallel intervals form the sides and the base, which also doubles as the height.
To determine the area under the function, with in the interval of,, one can estimate the area by trapeziums over subintervals by finding the .
In the first attempt approximate an area the curve of the given function above, two trapeziums will be considered.
The height (h) in this case, is the difference between the X intervals
Therefore, for the two trapeziums, h = is determined by;
, Where by n is the number of trapeziums or intervals.
So, we know b is 1 and a is 0 comparing to the integrals of the function,
So now,
Giving us, as our height (h)
NB:, 0.5 is considered to be the height because the trapezoids are rotated which makes their base the height.
To be able to calculate the area of both trapeziums, we have to consider all the lengths required hence,
We will therefore have to find the values of Y intervals, which form sides.
Middle
If we consider an area of under a curve on the interval
The interval could be divided into subintervals using
And then construct trapeziums over each sub interval,
The sum of the trapezoids constructed will probably give an approximation of an area under the curve on the interval
The new base (h) will therefore be,
And the sides will be,
Having found all the sides, and which also doubles as the base or height, we can now approximate the areas of each trapezium.
Therefore area will be as follows
Therefore, total approximation area for trapeziums
Comment; having estimated the area using 2, 5 and 10 trapeziums respectively, it can be noticed that the more trapeziums one uses to draw a conclusion, the nearer or the better approximation one gets. I.e. the more trapeziums one uses to approximate an area under a curve, the nearer you come to the solution got when u use the function to determine the area.
For instance, when two trapeziums are considered, the approximation is which makes a big difference from the area derived when using the function.
Also when 5 trapeziums are considered, the approximation is which also gives a greater difference of 1.002
But then it can be proved that the greater number of trapeziums one considers approximating the area under a curve, the more accurate answer you arrive at.
Conclusion
For the function ,
we can split the area covered under the curve formed by the function to estimate its area.
Y intervals, which form the other sides of the formed trapeziums, are crucial in estimating the trapezium areas. They form what in the formulae is
Knowing all the necessary sidesand theintervals, its then possible to estimate the area of every trapezium, the results give us, by applying the sides to the formulae of finding the area of trapeziums where in our case we consider;
Therefore, approximation for the areas of trapeziums gives the following results.
Therefore total approximation area will be;
This gives us the an approximation of
For the function , knowing the height which the sides will be,
The same procedure can apply to the above functions and other function as well to estimate their areas using trapezoids are the sides keep on varying depending on the structure of the curve formed, we always have to find news sides to that function but supposing the functions are in the same range.
.
The area estimated areas are as follows;
Areas
The total approximation of the areas will therefore be;
For the function
The y intervals of the new function will then be;
Having known all the required sides its then possible to estimate the areas of all the eight trapeziums formed under the curve.
Therefore the areas,
Which gives a total approximation area
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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