TABLE OF CONTENTS 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.
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.
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...
