Math IB SL Shady Areas Portfolio

This portfolio is an attempt at deriving and examining the scope and limitations of a general statement that can approximate the area under a curve using trapezoids. Generally, calculus – specifically the method of integration, is used to find the exact area under a curve. Although this method will be explored in comparison later in the portfolio, this investigation deals mainly with investigating a method to approximate the area using high school level math. First, this portfolio will attempt to derive a general statement that will give an approximation of the area under the curve of any function in any closed interval using trapezoids. Then, by applying the formula to sample functions, the answers given can be compared to the integral answers, allowing an examination into the accuracy of the trapezoid method of approximation. Lastly, by examining different behaviors of a graph, this portfolio will investigate the cause of any inaccuracies in this method.

The graph:  is given:

Figure 1 – Graph of

Using two trapezoids mapped onto the curve in the domain , the area under the curve in that domain can be approximated as the sum of the areas of the two trapezoids.

In order to map the trapezoids onto the graph, one must first divide the domain by the number of trapezoids being used, in order to find the height of the trapezoids (which is equivalent to each other). It should be noted that the height is not a vertical distance in this case, but the distance between the two parallel sides of a trapezoid. With the general case , height can be calculated with the formula    where  is the number of trapezoids.

In this case, the height will work out to:

.

Thus, the first (from the left) trapezoid will be mapped from  to , and the second trapezoid from  to . The first trapezoid will have vertices atand the second trapezoid will have vertices at :

Figure 2 – Two trapezoids mapped onto

The formula to calculate the area of a trapezoid is given as  where  is the height, and and  are the lengths of the parallel sides.

For the first trapezoid, the area is calculated to be:

A1

And the second trapezoid:

A2

And the total area being:

Atotal   A1 + A2

For the same function, examine the calculation when the same method is attempted with 5 trapezoids:

Figure 3 – Five trapezoids mapped onto

The height will work out to:

Thus, the trapezoids will be mapped between  and with equivalent heights of 0.2.

Using the formula for area of a trapezoid, the total area is calculates as the sum of 5 separate area calculations, one for each trapezoid. They all share the common factor of  , what  differentiates them will be the g(x) values. The first trapezoid will have those values as [g(0) + g(0.2)], the second as [g(0.2) + g(0.4)], and so on until the 5th trapezoid as [g(0.8) + g(1)]. This gives the following calculation:

Factoring out    and:

The area approximated by using 5 trapezoids differs in value to the approximation using only 2 trapezoids. Examine the following diagrams of the approximated area using 1-8 trapezoids:

Diagram 1 – 1-10 trapezoids mapped onto

In a chart, the approximated areas (calculated using the program Graphmatica) are:

Diagram 2 – Approximated Areas (left unrounded to showcase increasingly gradual differences)

One can notice with this data that as the area is approximated with an increasing number of trapezoids, the approximated area approaches a limit similar to how a function would approach an asymptote.

Comparing the calculation used to approximate the area with 5 trapezoids to the calculation ...