##### From Ship Stability for Masters and Mates, Sixth edition – Consolidated 2006

## Areas and Volumes

Simpson s Rules may be used to find the areas and volumes of irregular figures. The rules are based on the assumption that the boundaries of such figures are curves which follow a definite mathematical law. When applied to ships they give a good approximation of areas and volumes. The accuracy of the answers obtained will depend upon the spacing of the ordinates and upon how near the curve follows the law.

### Simpson s First Rule

This rule assumes that the curve is a parabola of the second order. A parabola of the second order is one whose equation, referred to co-ordinate axes, is of the form y = a _{0} + a _{1}x + a _{2}x ^{2}, where a _{0}, a _{1} and a _{2} are constants.

Let the curve in Figure 10.1 be a parabola of the second order. Let y _{1}, y _{2} and y _{3} be three ordinates equally spaced at h units apart.

Fig. 10.1

The area of the elementary strip is y dx. Then the area enclosed by the curve and the axes of reference is given by:

But

Using the equation of the curve and substituting x for O, h and 2h respectively:

Equating coefficients:

From which:

This is *Simpson s First Rule.*

It should be noted that Simpson s First Rule can also be used to find the area under a curve of the third order, i.e., a curve whose equation, referred to...

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