Sick of finding the area under graphs by counting squares? Seriously, counting squares is for kids. We can do better than that.
Finding areas underneath graphs has many uses; such as the finding displacement from velocity-time graphs and find work from force-position time graphs. So we’d like to have even more efficient techniques to do so at our disposal.
Then I got good news for you, because I’m going to go through a brutally efficient method to find the area under graphs, called the Trapezoid Rule. Trapezoid Rule is really handy at in situations where the graph y=f(x) cannot be integrated directly or the equation of the graph is unknown. Some examples of equation that cannot be integrated directly are:
But first, given the following trapezium with a perpendicular base, length on each side to be a and b, and width of h:
We can easily find the area to be:
So what has this got to do with finding the area under graph? Say we have a graph of y=f(x) and we'd like to find the area between x=a and x=b as shown:
The trick is to divide the interval into n equal parts with width of Δx, and draw trapeziums underneath the graph, with heights reflecting the values of f(x):
Now applying the formula for the area of a trapezoid as shown earlier, the area under the graph of y=f(x) between x=a and x=b can be approximated by find the total area of the trapezoids, which can be shown to be:
Now we can see the importance of diving the interval equally. It is done so that Δx can be factorized out and the values of f(x) can be combined together nicely.
The application of this formula is very straightforward by the following steps:
Let's use the Trapezoid Rule to approximate the area underneath the graph y=x² between x=0 and x=2, and compare to the actual value.
We are going to divide by 4 parts, and choose Δx = 0.5, so the interval between 0 to 2 will be divided into 0, 0.5, 1.0, 1.5 and 2.0 as shown:
The area can then be approximated to be:
Meanwhile, the actual area can be found to be 2.666... (8 over 3) by using definite integrals.
The use of the trapezoid rule slightly overestimated the actual area by only 3%! Not too bad considering we only divided the interval into 4 parts and we managed to approximate the area without the use of integration.
Of course the approximation can be further improved by using smaller size for Δx, but that will make the calculation more tedious, so that's best left for the computer.
Now armed with the Trapezoid Rule, go and amaze your friends who are still trying to count squares underneath graphs!