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Shady areas; math portfolio type 1

Extracts from this document...

Introduction

Shady areas

This investigation is carried out in order to find a rule to approximate the area under a curve using trapeziums (trapezoids).

From calculus it is known that by the help of integration, the definite area under a curve could be calculated. In this case a geometrical method will be used in order to approximate the area under a curve.

The function g(x) = x2+3 is considered. The graph of g is shown below. The area under the curve from x = 0 to x = 1 is approximated by the sum of the area of two trapeziums.

image00.png

In order to approximate the area under this curve, the composite trapezoidal rule is used. The composite trapezoidal rule is a numerical approach for approximating a definite integral.

image01.pngg(x) dx = Area

image01.pngx2 + 3 dx = Area

According to composite trapezoidal rule:

Area = ½ (width of strip) [first height + 2(sum of all middle heights) + last height]

Since there are two trapeziums from 0 to 1 in fig 1.1 hence the width of each strip: 1/2 = 0.5

x

g(x) or height

0

02 + 3 = 3

1/2

(1/2)2 + 3 = 3.25

1

12 + 3 = 4

So: Area = ½ (0.5) [3 + 2( 3.25) + 4]

     Area = 3.37 area units (a.u.)

Increasing the number of trapeziums to five:

image17.png

Approximating the area in this case with five trapeziums:

Width of every strip: 1/5 = 0.2

x

g(x) or height

0

3

1/5

3.04

2/5

3.16

3/5

3.36

4/5

3.64

1

4

Area = ½ (1/5) [3 + 2(3.04 + 3.16 + 3.36 + 3.64) + 4]

Area = 3.34 a.u.

...read more.

Middle

3.34

1.3

3.3359375

1.4

3.334490741

1.5

3.333984375

It is clearly observed that as the number of trapeziums increase, the uncertainty in approximating the area decreases since increasing number of segments of area under the curve are being taken into account. Thereof the value gets closer and closer to the actual value of the integral which is: 3.333333333 (from integration).

In order to find a general expression for the area under the curve of g(x) = x2 + 3 from x = 0 to x = 1, the following diagram is considered with n trapeziums:

image21.png

As mentioned earlier, area according to composite trapezoidal rule is:

Area = ½ (width of strips) [first height + 2(sum of all middle heights) + last height]

So the general expression in this case for n trapeziums is:

Area = ½ (x) [g (0) + 2(g (x1) + g (x2) +…………….+ g (xn-1)) + g (1)]

[OBS! Width of every strip is x. g (0) = g (x0) and g (1) = g (xn)

Using the results, the general statement that will estimate the area under any curve y = f (x), from x = a  to x = b, using n trapeziums could be developed.

Assuming that the following graph is y = f (x):

image22.png

Composite trapezoidal rule: Area = ½ (width of strips) [first height + 2(sum of all middle heights) + last height]

So the general statement for approximating the area under any curve according to fig 1.7:

Area = ½ (c) [f (a) + 2( f (x+c) + f (x+2c) +……………..+ f (x+(nc-1)) + f (b)]

[OBS! Width of every strip is c. f (a) = f (x) since (a = x) and f (b) = f (x+nc) since (b = x+nc)]

Using the general statement with eight trapeziums the area under the following three curves from x = 1 x = 3 could be found:

  1. y= image03.png

image02.png

  1. y= 4x3 – 23x2 + 40x – 18
  1. y= image03.png

image04.png

Area = ½ (c) [f (a) + 2( f (x+c) + f (x+2c) +……………..+ f (x+(nc-1)) + f (b)]

Where: c = 2/8 = 1/4 = 0.25, a = 1 and b= 3 in this case. [a = x and b = (x +nc) from fig 1.7]

x values

f(x)

1

0.6299605249

1+1/4 = 5/4

0.7310044346

1 +2( 1/4) = 3/2

0.8254818122

1 +3( 1/4) = 7/4

0.9148264275

1 +4( 1/4) = 2

1

1 +5( 1/4) = 9/4

1.081687178

1 +6( 1/4) = 5/2

1.160397208

1+7( 1/4)  = 11/4

1.236521861

1 +8( 1/4) = 3

1.310370697

Area = ½ (1/4) [0.6299605249 +2 (0.7310044346 + 0.8254818122 + 0.9148264275 + 1 + 1.081687178 + 1.160397208 + 1.236521861) + 1.310370697]

Area = 1.980021133 a.u.

image02.png

image05.png

Area = ½ (c) [f (a) + 2(f (x+c) + f (x+2c) +……………..+ f (x+(nc-1)) + f (b)]

Where: c = 2/8 = 1/4 = 0.25, a = 1 and b= 3 in this case. [a = x and b = (x +nc) from fig 1.7]

x values

f(x)

1

2.846049894

1+1/4 = 5/4

3.474792043

1 +2( 1/4) = 3/2

3.837612894

1 +3( 1/4) = 7/4

4.156356486

1 +4( 1/4) = 2

4.365641251

1 +5( 1/4) = 9/4

4.484455912

1 +6( 1/4) = 5/2

4.534134497

1+7( 1/4)  = 11/4

4.534086944

1 +8( 1/4) = 3

4.5

Area = ½ (1/4) [2.846049894 + 2(3.474792043 + 3.837612894 + 4.156356486 + 4.365641251 + 4.484455912 + 4.534134497 + 4.534086944) + 4.5]

Area = 8.265026244 a.u.

  1. y = 4x3 – 23x2 + 40x – 18

image06.png

Area = ½ (c) [f (a) + 2(f (x+c) + f (x+2c) +……………..+ f (x+(nc-1)) + f (b)]

Where: c = 2/8 = 1/4 = 0.25, a = 1 and b= 3 in this case. [a = x and b = (x +nc) from fig 1.7]

x values

f(x)

1

3

1+1/4 = 5/4

3.875

1 +2( 1/4) = 3/2

3.75

1 +3( 1/4) = 7/4

3

1 +4( 1/4) = 2

2

1 +5( 1/4) = 9/4

1.125

1 +6( 1/4) = 5/2

0.75

1+7( 1/4)  = 11/4

1.25

1 +8( 1/4) = 3

3

...read more.

Conclusion

fig 3.1), for instance, is more straight and geometrical compared to the graph of a quadratic equation(fig3.2) and the graph of latter is more geometrical compared to the graph of a fifth degree polynomial(fig 3.3). Hence the shape of a graph influences the approximation to a great extent and therefore the more geometrical and linear the shape of a graph the more accurate will be the approximation. The relationship between the shape of a graph and the accuracy of approximation is also evident in figures: 2.1, 2.2, 2.3 and the data related to them. The accuracy decreases in b (fig 2.2) compared to a (fig 2.1) as the shape of the graph becomes less linear and more curved. Similarly the accuracy decreases in c (fig 2.3) compared to b (fig 2.2) as the shape of the graph becomes more haphazard and less linear.

The accuracy of approximation also depends upon the number of trapeziums used. This is evident in figures (1.1 – 1.7) and the data related to them. As the number of trapeziums increases, the approximate value gets closer and closer to the true value of the area.

It could be concluded that the composite trapezoidal rule is exact for polynomials of degree less than or equal to one.

Math Portfolio SL type 1

Shiba Younus IB 2

                                                                                Date: 23-04-09

                                                                                Katedralskolan

...read more.

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

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