AREA OF IRREGULAR SHAPES: HISTORY OF TRAPEZIUM RULE

Theory & method

The Trapezium Rule is a method of finding the approximate value of an integral between two limits.

The area involved is divided up into a number of parallel strips of equal width.

Each area is considered to be a trapezium(trapezoid).

If there are n vertical strips then there are n+1 vertical lines(ordinates) bounding them.

The limits of the integral are between a and b, and each vertical line has length y₁ y₂ y₃... y_n+1

The trapezium rule

The needs for approximation

There are some functions that cannot be integrated exactly (and some which one will not be able to integrate due to the level of knowledge that one has upon the topic of integration).

Therefore it is useful to be able to give an approximation, and say whether it is above or below what one might expect the real value to be. (In later study one might be able to predict the accuracy of these estimates).

The trapezium rule: simple form

Consider that there is a function of x: .

If one was to plot the graph , and then wanted to approximate the area under the curve between two bounds, one could use trapezia.

The idea of one trapezium is simple. Take the two bounds as the two vertical edges of the trapezium, and then take a line between them as the sloping line at the top. The x-axis is the width, and base line.

One can see that this is a rather crude approximation for larger areas, but consider the idea of multiple trapezia, this is much more accurate. The simple form of the trapezium rule refers to the single trapezium, and can be shown:

$\displaystyle \int _{a}^{b} f(x) \ dx \approx \frac{1}{2} (b - a) \left[ f(b) + f(a) \right]$

(The area of a trapezium is the sum of the parallel sides multiplied by the width, multiplied by a half).

This gives the most basic approximation, and can be improved (in most cases).

The trapezium rule: general form

One can consider that as one allows the number of trapezia to increase, and therefore their widths to decrease, one will have a better approximation of the value of the integral. One thing that one should note is that good setting-out of work is crucial, as sometimes one might be expected to deal with 8 trapezia, and therefore the working can become incomprehensible if it is not properly organised.

Consider now that between the bounds, a, and b, one has an indefinite number of trapezia, denoted:

$x_{0}, \ x_{1}, \ x_{2}, \ ... \ , \ x_{n}$

Where:

$x_{0} = a$

$x_{n} = b$

Now one can begin the formal representation:

The one trapezium example is:

$\frac{1}{2} (x_{1} - x_{0}) \left[ f(x_{1}) + f(x_{0}) \right]$

Hence, one can use sigma notation to show the total area:

$\displaystyle \sum _{n = 1}^{n} \left{ \frac{1}{2} (x_{n} - x_{n - 1}) \left[ f(x_{n}) + f(x_{n - 1}) \right] \right}$

Hence:

$\displaystyle \int _{a}^{b} f(x) \ dx \approx \sum _{n = 1}^{n} \left{ \frac{1}{2} (x_{n} - x_{n - 1}) \left[ f(x_{n}) + f(x_{n - 1}) \right] \right}$

HISTORY OF TRAPEZIUM RULE

The trapezium rule

The needs for approximation

The trapezium rule: simple form

The trapezium rule: general form

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