# 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

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