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 1x + a 2x 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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