Subject: Calculus

Trapezium Rule

Integral is one subject in Calculus that uses a lot of approximations and definite integral is the kind of integral that frequently abuses approximations in solving for the integral. One example of this is the trapezium rule. This rule employs a method of approximating an area under any given curve and since we know that an area under a curve is given by integration, so the trapezium rule gives a method of estimating integrals.

Why approximate integrals?

As far as Mathematics is concerned, it is somewhat our responsibility to keep track of the accuracy of every mathematical entity. But if we are face with integrals that are very difficult to integrate directly, then we have no choice but to approximate. Even the integral itself is just an approximation!
Remember that trapezium rule is just an approximation (a very rough approximation) so it is better to use this technique rarely and only when the integral is very difficult to perform directly and when all else fails.

How it works?

As mentioned earlier, integration is just a mere representation of an area of a function. So if we know the area, then doing integration is not anymore a necessity. Suppose a function f is integrated from point a to point b, then its integral is just the area below the curve of f from point a to point b (see figure at the right). Calculating that area under the curve is not easy! That's where trapezium rule comes in because in this method, all we have to do is to calculate the trapezoidal area (the green area) associated with the curve which is at most times very easy and straightforward to calculate.

From our geometry class, we know already how to calculate the area of a trapezoid. So for the figure at the left with respective heights given by f(a) and f(b), the area of the trapezoid (green region) at the left is given by

A = \frac{1}{2}\left (f(a) + f(b)\right )(b-1)

And since we already have the value of the area, the approximate value of the integral of the function f evaluated from a to b can be approximated as

\int_{a}^{b}f(x)dx \approx A = \frac{1}{2}\left (f(a) + f(b)\right )(b-1)

Avoiding big errors

To avoid large errors in computing integrals using trapezium method, we must divide the region under the curve into smaller vertical trapezoids and then sum up the area of that smaller trapezoids. Of course this is tedious because we have to solve many trapezoidal areas but its the best thing to do when we have no room for large errors.

This techniques helps reduce errors because as you can see at the figure above, the "tips" of each trapezoid almost religiously follow the curve of the function thus making the sum of the area closer to the actual area of the region below the curve.

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