• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18

Area under curve

Extracts from this document...

Introduction

In this investigation we will be estimating the area g(x) between X=0, X=1, this area is approximate by the sum of the two trapeziums’. Let one go 0→0.5 and the other 0.5→1. The area of a trapezium is ½ (a+b)h will be used to identify the total area throughout all the questions.

image00.png

image01.png

A₁ +A₂=(½ (3+3.25) x 0.5) + (½ (3.25+4) x 0.5)

= 1.56 + 1.81

= 3.37

image17.pngimage17.png = 0.2 here each trapezium will be 0.2 of each area, given that g (0) =3, g (0.2) = 3.04.... These values can be seen on the graph below. This area is approximate by the sum of the 5 trapeziums.

image23.png

image26.png

A₁ +A₂+A3+A4+A5 = (½ (3+3.04) x 0.2) + (½ (3.04+3.16) x 0.2) + (½ (3.16+3.36) x 0.2) +       (½ (3.36+3.64) x 0.2) + (½ (3.36+4) x 0.2)

= 0.60 + 0.62 + 0.65 + 0.70 + 0.76

= 3.34

With the help of GDC, we investigate what happens as the number of trapeziums increase:

image32.png

image37.png

A₁ + A₂ + A3 + A4 + A5 + A6 + A7 + A8 =(½ (3+3.015) x 0.125) + (½ (3.015+3.062) x 0.125) + (½ (3.062+3.14) x 0.125) + (½ (3.14+3.25) x 0.125) + (½ (3.25+3.390) x 0.125) +             (½ (3.390+3.562) x 0.125) + (½ (3.56+3.765) x 0.125) + (½ (3.765+3.4) x 0.125) =

= 0.375 + 0.379 + 0.389 + 0.399 + 0.415 + 0.434 + 0.457 + 0.485

= 3.333

image43.png

image48.png

A₁ + A₂ + A3 + A4 + A5 + A6 + A7 + A8 +A9 + A10 =(½ (3+3.01) x 0.1) + (½ (3.01+3.04) x 0.1) + (½ (3.04+3.09) x 0.1) + (½ (3.09+3.16) x 0.1) + (½ (3.16+3.25) x 0.1) +                                (½ (3.25+3.36) x 0.1) + (½ (3.36+3.49) x 0.1) + (½ (3.49+3.64) x 0.1) +                                   (½ (3.64+3.81) x 0.1) + (½ (3.81+4) x 0.1) =

= 0.3005 + 0.3025 + 0.3065 + 0.3125 + 0.3205 + 0.3305 + 0.3425 + 0.3565 + 0.3725 + 0.3905

= 3.335         

(4 significant figures were used to demonstrate the difference between n= 8, n= 10)

image02.png

image09.png

A₁ + A₂ + A3 + A4 + A5 + A6 + A7 + A8 +A9 + A10 + A11 + A12 + A13 + A14 + A15 + A16 + A17 + A18 + A19 + A20=

(½ (3+3.002) x 0.05) + (½ (3.002+3.01) x 0.05) + (½ (3.01+3.02) x 0.05) +

(½ (3.02+3.04) x 0.05) + (½ (3.04+3.06) x 0.05) + (½ (3.06+3.09) x 0.1) +

(½ (3.09+3.12) x 0.05) + (½ (3.12+3.16) x 0.05) + (½ (3.16+3.20) x 0.05) +

(½ (3.20+3.25) x 0.05) + (½ (3.25+3.30) x 0.05) + (½ (3.30+3.36)

...read more.

Middle

 (g1 + g2)image14.pngimage14.png + image19.pngimage19.png(g2 + g3) +..................................+ image19.pngimage19.png(gn-1 + gn)image14.pngimage14.png.

= image20.pngimage20.png((g1 + g2) + (g2 + g3) + (g3 + g4) + ...........................................+ (gn-1 + gn)).

= image20.pngimage20.png(g0 + gn + 2g1 + 2g2 + 2g3 + ..........................................+2gn-1 +gn).

Total Area f(x) dx

image21.pngimage21.png dx image20.pngimage20.png(g0 + gn + 2(g1 + g2 + g3 + .......................................+gn-1)).  Equation [1]

Let c (g1 + g2 + g3 + ..........................................+gn-1).

C is the sum from n=1 using sigma summation notation.

C = image22.pngimage22.png

image24.pngimage24.png dx image20.pngimage20.png(g0 + gn + 2image22.pngimage22.png).

From the general case for g(x), we will be calculating the area under the curve for f (x) from x=b to x=a using n trapeziums.

image25.png

From the diagram above:

D = b –a

D = nh

h = image13.pngimage13.png

Using the following notation:

f(a) = f0

f(a + h) = f1

f(a+2h) = f2

image27.png

f(a+(n -1)h) = fn – 1

f(b) = fn

This follows a similar derivation as for g(x) from 1 to 0.

image28.png

The following formula comes from equation [1] above, this can be used because the derivation for g(x) and f(x) is similar, the only difference being instead of saying b-a=1 we just use b-a in the expression.

image29.pngimage29.png dx image30.pngimage30.png(f0 + fn + 2f1 + 2f2 + ..........................................+2fn-1).

(f0 + fn + 2(f1 + f2 + ..........................................

...read more.

Conclusion

GENERAL STATEMENT:

  • The general statement accurately represents the area under the curve from b to a, if
  • The curve doesn’t change rapidly with x, for example the following two curves demonstrate this for n=2.
  • The larger the area you are examining the larger the value of n has to be in general
  • [basically b-a/2n has to be a small value, in the examples above the value of 1/8 gave an error of about .5%, therefore it would be expected that if b-a/2n was of about this magnitude the trapezium rule would give an accurate fit of the area under the curve from b to a.
  • Significant figures, because there is a difference in values, there wasn’t a set number of significant figures.
  • The general statement doesn’t work, if b-a/2n is large
  • For functions that change periodically, i.e. sin (x), and if you take trapeziums that are larger than one period of the function.
  • If the function is negative in some areas, and these area are ignored by the edges of the trapeziums.

Overall the trapezium does provide a good fit to the area under the curve from b-a if the above conditions are met.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Shady Areas. In this investigation you will attempt to find a rule to approximate ...

    in units squared AC is the approximated area for trapezium C, in units squared AD is the approximated area for trapezium D, in units squared AE is the approximated area for trapezium E, in units squared Again an assumption must be made in regards to the length of the bases for each of the individual trapeziums.

  2. Moss's Egg. Task -1- Find the area of the shaded region inside the two ...

    to its base (D). Therefore, triangle ABC is an isosceles triangle. Thus we see that point C extending directly below to the base triangle ABC is the radius of the small circle, which equals 3 cm. Therefore, to calculate the area, we simply substitute the information we have to the

  1. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    In Graph 1.3 has been flipped over and stretched by a factor of to give the new graph of . Overall, it can be seen that in the represents the 'height' of the graph. In mathematics this 'height' is known as the amplitude of the graph, meaning that in the equation above, represents the amplitude of the graph.

  2. Investigating the Ratios of Areas and Volumes around a Curve

    seems to hold for the new limits. Any proof of the correlation must therefore explain the instances where it seems not to hold. To test further instances I produced a program on my Casio fx-9750G+ graphing calculator. The following program performs the above process automatically: ?�N ?�A ?�B ?(Y^(1/N),A^N,B^N)/?(X^N,A,B)

  1. Investigating Sin Functions

    Now, from these graphs, one would likely assume that as the Amplitude of the graph increases, the shape vertically stretches, and as the Amplitude decreases, the graph shrinks. However, one must consider possible limitations to this statement. Let me pose a question, what of negative numbers??

  2. Infinite Summation

    Profilex..."MHa�����-��$T& R�+S�e�L b�}w�g�(tm)�-E""�u�.VD��N�C�:D(tm)u� �E^"��;""cT3/403�y���|1/2�U�R�cE4`��"�ÞvztL�U�F\)�s:��(c)��k�-iYj"��6|"v(tm)P4*wd>,y<��'/�<5g$(c)4�!7�C�N�-��l��C��T�S"3-q";�-E#+c> �vڴ��=�S԰��79 ڸ�@�-`Ó�m�-�v�Ul�5��`�P�=��G���j��)�k�P*}�6�~^/�~�.�~�a ���2 n�ײ0�%��f������|U ��9�l��7?���j`�'�l7���"�t�i��N�f]?�u�h...�gM Zʲ4��i(r)�"[�&LY��_�x� {x�O��$1/4�̥߬S]�%��֧���&7��gÌ>r=��*g8`�(tm)� 8rʶ�<(c)�����"d�WT'"�<� eL�~.u"A(r)�=9(tm)�-�]��>31�3'�X3����-$e�}��u,��gm'g...6�64$Ñ�E zL*LZ�_�j���_��]1/2X��y�[�?...Xs ���N��/� �]��|m�3/4�(tm)sÏ�"k_Wf-�ȸA�23/4�)�o�� z-di�������2�|m٣��j|5Ô¥ej�8��(r)e�E��7��[���Q�|�IM%ײ�xf)�|6\ k��"`Ò²"�ä.<k��U�}j�� M=��"mjß��Ü�� ��e�)�`c���IWf����/���^a �44�M���i ��6p�"��_� ޡ��/IDATH �-;N1��R$^�+*�()(hi(c)(\Q�Ss.��Q���p��A..."��6-�k�:N�A�-f�e��3���z� ��3/4�kA�_��� �lI���8�ι?s�i[3/4/�"���...�_E#Ñ����'��_��b ���x���� :u-1/4N� T-��Yp�"�(c)"Fg(�� h��=��h�' -tx�Y�.�SPQ* 5�Y���#A���vE���˭����k-�BRi7�dc��TI�|�=Q<"&1/4!�3/4�E"�'#q)x18�M�qH[ {� S-�^������B�"iÓn�[1/4(tm)b��ys�!� ��o� ~!z1/2�H�Rt/jÍ m��-l����"��Q�Y�;\Æ ï¿½q'�ul|^�b2�v8:c=�Mb�Z�1q(r)��' 1 �93q"$¥ï¿½( ���$�3�A����qY8g�O��"�.6�[email protected]�"[�ף�$G��c���n�R(tm)� �O,�go�(tm)-Y\rM�=C�(1/4�-�#0Hv�"�R�(r)T�'H $9�-��c���Kc�^QAr���*�3C�iZq���e��4i{�X�IEND(r)B`�PK!("�� (word/embeddings/Microsoft_Equation40.bin�p^�ÂRÐ3ÿ�� lH�@6������?X H(tm)�`...�_ [A� �C��N�B�2.(tm)4+<σ��s '���dp�P�Éi�ä§[email protected]%�rR�<�).`)"

  1. The investigation given asks for the attempt in finding a rule which allows us ...

    However, more calculations will reveal a more accurate answer. Twenty Trapezoids Trapezoid h a b Area 1 0.05 3 3.0025 0.1500625 2 0.05 3.0025 3.01 0.1503125 3 0.05 3.01 3.0225 0.1508125 4 0.05 3.0225 3.04 0.1515625 5 0.05 3.04 3.0625 0.1525625 6 0.05 3.0625 3.09 0.1538125 7 0.05 3.09 3.1225

  2. In this investigation, I will be modeling the revenue (income) that a firm can ...

    P=-0.75Q+6 3. R=(-0.75Q+6)(Q) Quarter Two: 1. Revenue Equation: R=PQ= 2. P=-1.25Q+4.5 3. R=(-1.25Q+4.5)(Q) Quarter Three: 1. Revenue Equation: R=PQ 2. P=-2Q+4 3. R=(-2Q+4)(Q) Quarter Four: P=-Q+6 1. Revenue Equation: R=PQ 2. P=-Q+6 3. R=(-Q+6)(Q) Now since we have our revenue functions (seen as quadratic functions) we can graph the functions to investigate whether the results make sense and seeing the benefits and limitations to the VBGC.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work