Basic Technical Mathematics, 11th Edition

2.5 Measurement of Irregular Areas

• Trapezoidal Rule • Simpson’s Rule

In practice it may be necessary to find the area of a figure with an irregular perimeter or one for which there is no specific formula. We now show two methods of finding a good approximation of such an area. These methods are particularly useful in technical areas such as surveying, architecture, mechanical design.

THE TRAPEZOIDAL RULE

For the area in Fig. 2.98, we draw parallel lines at n equal intervals between the edges to form adjacent trapezoids. The sum of the areas of these trapezoids, all of equal height h, is a good approximation of the area. Now, labeling the lengths of the parallel lines $y_0\hspace{0.17em},\hspace{0.17em}{y}_1\hspace{0.17em},\hspace{0.17em}{y}_2\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em},\hspace{0.17em}{y}_n\hspace{0.17em},\hspace{0.17em}$ the total area of all trapezoids is

$$\begin{array}{l}\begin{array}{l}\begin{array}{c}\text{first}\\ \text{trapezoid ...}\end{array}\end{array}\end{array}$$