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type 1 maths portfolio trapezium rule

Extracts from this document...

Introduction

image00.pngThere 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;

image01.png Or in words, the average of twoimage00.pngsides times the base, which could also be expressed as image141.png where by if considering trapeziums formed under a curve; parallel image48.pngintervals form the sides and image22.png the base, which also doubles as the height.

To determine the area under the functionimage24.png, with in the interval ofimage168.png,image177.png, one can estimate the area by trapeziums over subintervals by finding the image22.png.image00.pngimage12.png

In the first attempt approximate an area the curve of the given function above, two trapeziums will be considered.

image23.pngimage31.png

The height (h) in this case, is the difference between the X intervalsimage41.png

Therefore, for the two trapeziums, h =image00.pngimage41.png is determined by;

image72.png, 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,

image80.png

                                 So now, image72.png

image95.png

image104.png

            Giving usimage113.png, as our height (h)

NB:image41.png, 0.5 is considered to be the height because the trapezoids are rotated image133.png 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, image136.png

We will therefore have to find the values of Y intervals, which form image137.pngsides.

...read more.

Middle

with corresponding ordinatesimage175.png.

If we consider an area of under a curve on the interval image176.png

The interval could be divided into subintervals using image13.png

image178.png

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 image00.pngimage179.png

The new base (h) will therefore be,

image47.pngimage180.pngimage181.png

And the sides will be,

image182.pngimage183.pngimage184.pngimage185.pngimage186.png

image187.pngimage188.pngimage189.pngimage190.pngimage191.pngimage192.png

image00.png

  Having found all the sides,image48.png and image22.pngwhich also doubles as the base or height, we can now approximate the areas of each trapezium.

Therefore area will be as follows

image02.pngimage03.pngimage04.pngimage05.png

image06.pngimage07.pngimage08.png

image09.pngimage10.pngimage11.png

Therefore, total approximation area for image13.pngtrapeziums

image14.png

image15.pngimage16.png

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 image17.png which makes a big difference from the area derived when using the function.

Also when 5 trapeziums are considered, the approximation is image18.png 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.

...read more.

Conclusion

image81.pngand image82.png are put in the given functions instead for image21.png will give different results.

For the function image83.png,

image84.png

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 image74.png

image85.pngimage86.pngimage87.pngimage88.pngimage89.png

image90.pngimage91.pngimage92.pngimage93.png

Knowing all the necessary sidesimage22.pngand theimage48.pngintervals, 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 image94.png where in our case we consider; image96.png

Therefore, approximation for the areas of image73.png trapeziums gives the following results.

image97.pngimage98.pngimage99.pngimage100.png

Therefore total approximation area will be; image101.png

This gives us the an approximation of image102.png

For the function image103.png, knowing the height whichimage22.png the sides will be,

image105.png

The same procedure can apply to the above functions and other function as well to estimate their areas using trapezoids are the image74.pngsides keep on varying depending on the structure of the curve formed, we always have to find news sides to that function but image22.pngsupposing the functions are in the same range.

.

image106.pngimage107.pngimage108.pngimage109.png

image110.pngimage111.png

The area estimated areas are as follows;

Areas

image112.pngimage114.pngimage115.png

image116.pngimage117.pngimage118.png

The total approximation of the areas will therefore be;

image119.png

image120.png

image121.png

For the function image122.png

image123.png

The y intervals of the new function will then be;

image124.pngimage125.pngimage126.pngimage127.pngimage128.png

Having known all the required sides image22.pngimage129.pngits then possible to estimate the areas of all the eight trapeziums formed under the curve.

Therefore the areas,

image130.pngimage131.pngimage132.pngimage134.png

Which gives a total approximation area image119.png

image135.png

...read more.

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