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Maths Portfolio Shady Areas

Extracts from this document...

Introduction

Shady Areas

Standard Level

Name Anis Mebarek

Candidate Number:

Shady areas:

We are able to find the area under the graph through calculating the area of the trapeziums under it. Through having numerous trapeziums under the curve the amount of uncertainty within the answer decreases this is due to the uncalculated area decreasing. I will be investigating and proving this theory from using the function image19.pngimage19.png and to find out the area of the trapezium I will use this formula image56.pngimage56.png. image18.jpg

image00.png

As seen from the graph the area between the curve and the trapezium is also being calculated when trying to find the area under the line image75.pngimage75.png therefore we are expected to get a higher number then the area it’s self. But increasing the number of trapeziums on the graph from 1 to 2, the uncertainty for area decreases.  This can be seen from the graph below.

image01.pngimage43.png

Therefore through increasing the number of trapeziums within the graph the uncertainty of the answer will decreases and so will the number for the area will decrease, as the trapeziums will over-estimate the number for the area. To prove this I will have to calculate the actual area under the curve through using integration.

image90.png

image96.png

image20.png

image29.png

...read more.

Middle

T5

3.34

I have noticed there is a relationship between the number of trapeziums within the graph and the areas that they produce. The table below will show the differences in the areas that each number of trapeziums produces.

Number of trapeziums

Total area

Total difference

T1

3.5

0

T2

3.37

0.125

T4

3.34375

0.03125

T5

3.34

0.00375

When we take the total difference and square it, then add the two of the same squared differences.

image67.png

image68.png

image69.png

image70.png

Therefore by finding the potential “total difference” and have the total area be subtracted from the found total difference, we will be able to find the next total area.

Part 2

So the very basic idea behind finding the area underneath the line is through finding to how many trapeziums to separate the X axis into, which is deciding upon how much height each section will have. Then finding the trapeziums area through doing the image71.pngimage71.png. Therefore for example when we are trying to find the area with 5 trapeziums the formula will be:

image72.png

But when we have this equation in terms of x we get:

image73.png

Therefore it’s basically just plugging in the numbers into the equation above, for example for image74.pngimage74.png you will have to find that length therefore you plug the 0.2 into the given equation, for this example it’s,
image45.png
when we find the number of that X co-ordinate we can then plug it into the equation, and repeat the process with the other lengths. But his is not a general formula, where I will have to change the image76.pngimage76.png to image77.pngimage77.png and the different x axis sides, where the image77.pngimage77.png will be used, which will give me this formula:

image78.png
To simplify this equation I will simplify one part of it as it correlates towards the rest, as to use that part as an example:

image79.png

Therefore we are able to factorize this equation as ½ and h as repeated throughout the equation which will give us this:

image80.png


This is still not a general explanation to the formula, as its still focused on one example, therefore to change this example so that it can be used with any number of trapeziums, we have to change the height as it has to correlate with the length of the x-axis and the number of times that x-axis will be split. Simply we are able to change h to image81.pngimage81.png where “b-a” will give us that length of the x-axis where they want to measure the area from, essentially the upper and the lower limits of their search for the area. “n” in that equation is the number of trapeziums that we want to use. Therefore together, “b-a” will be split by the number of trapeziums someone wants to use, therefore giving us height for the trapezium.

image82.png

We can further simplify this equation into:

image83.png

We are able to do this because, the numbers repeat twice throughout the equation apart from image84.pngimage84.pngand image85.pngimage85.png though by doing so, we are just making it simpler to use, but to make it more general we will have to replace image84.pngimage84.png with image86.pngimage86.pngtherefore transforming the equation into this:

image87.png

“…” in the formula is used as an uncertainty to how many trapeziums the user will have on his graph, therefore they have to continue depending adding until they reach the third last trapezium. Until reaching “n-1” where n is the number of trapeziums, this will allow the equation to substitute the uncertainty of the number of trapeziums within the equation. Though we are able to further simplify this equation to:

image88.png

As is was simply image89.pngimage89.png

Therefore the most general formula is:

image88.png

Consider the areas under the following three curves, from x=1 to x=3:

image22.png

image24.png

image26.png

image22.png

X

y

2g

1

0.629960525

-

1.25

0.731004435

1.462009

1.5

0.825481812

1.650964

1.75

0.914826428

1.829653

2

1

2

2.25

1.081687178

2.163374

2.5

1.160397208

2.320794

2.75

1.236521861

2.473044

3

1.310370697

-

...read more.

Conclusion

image28.pngimage28.png, selected the lower limit to be 1, and the upper limit to be 3, then pressed enter to find the area between the limits.

Area

Y=

Trapezium

Integration

Difference


image22.png

image30.png

image31.png

0.0007


image24.png

image32.png

image33.png

0.1335


image26.png

image34.png

image35.png

0.0205

The trapezium rule came to a close proximity with its approximations, with only small differences between the actual area, and the estimation. Though increasing the number of trapeziums used within the equation would increases the accuracy of the answers.

The scope and limitations:image36.png

image06.png

image37.png

The way that the rule will be limited or even not work is through the sin curves, as they fluctuate between the x-axis, the areas calculated will be difficult to do, as the sides of the graph go from a positive number, to a negative, therefore canceling each other out. image38.pngimage07.png

Another limitation to the rule is when the graph is irregular, and then the uncertainty or the over/underestimations from the rule will increase as the irregular curves create difficulty in estimating the curving parts of the graphs. Even through increasing the number of trapeziums within the graph the uncertainty is still great.image39.png

image08.png

Another limitation is an isotope, where the line does not touch the axis, therefore makes it difficult to calculate the lengths for the trapeziums. Which makes it impossible to do, but if it’s done then the uncertainty for the area found will be high.

image40.png

image09.png

...read more.

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