178 Chapter 7 Length and Length-Related Parameters
Substituting for the values of each trapezoid,
(7.5)
and simplifying Equation (7.5) leads to
(7.6)
Equation (7.6) is known as the trapezoidal rule. Also note that the accuracy of the area approx-
imation can be improved by using more trapezoids. This approach will reduce the value ofh
and thus improve the accuracy of the approximation.Counting the Squares There are other ways to approximate the surface areas of irregular shapes.
One such approach is to divide a given area into small squares of known size and then count the
number of squares. This approach is depicted in Figure 7.12. You then need to add to the areas
of the small squares the leftover areas, which you may approximate by the areas of small triangles.Subtracting Unwanted Areas Sometimes, it may be advantageous to first fit large primitive
area(s) around the unknown shape and then approximate and subtract the unwanted smaller
areas. An example of such a situation is shown in Figure 7.13. Also keep in mind that for
symmetrical areas you may make use of the symmetry of the shape. Approximate only 1/2, 1/4,
or 1/8 of the total area, and then multiply the answer by the appropriate factor.Aha
1
2
y 0 y 1 y 2 pyn 2 yn 1
1
2
ynb
A
h
2
1 y 0 y 12
h
2
1 y 1 y 22
h
2
1 y 2 y 32 p
h
2
1 yn 1 yn 2
yn
h
y 1 y 2 y 3
A 1 A 2 A 3
An
y 0
■Figure 7.11
Approximation of a planar area by
the trapezoidal rule.■Figure 7.12
Approximation of a planar area
using small squares.Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀