Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

7.6 Area 179

Weighing the Area Another approximation procedure requires the use of an accurate analyti-

cal balance from a chemistry lab. Assuming the profile of the area to be determined can be

drawn on a 8 11 sheet of paper, first weigh a blank 8 11 sheet of paper, then using the

analytical balance, weigh the sheet and record its weight. Next, draw the boundaries of the

unknown area on the blank paper, and then cut around the boundary of that area. Determine

the weight of the piece of paper that has the area drawn on it. Now, by comparing the weights

of the blank sheet of paper to the weight of the paper with the profile, you can determine the

area of the given profile. In using this approximation method, we assumed that the paper has

uniform thickness and density.

EXAMPLE 7.1 Using the trapezoidal rule, determine the total ground-contact area of the athletic shoe shown

in Figure 7.14. All dimensions are shown in inches.

1

2

1

2

■Figure 7.13

Approximation of a planar area

using a rectangular primitive and

small squares and triangles.

1.12 1.37 1.25 1.00 0.87 1.12 1.75 2.12 2.12 2.00 1.87

1.00

1.50 1.62 1.75 1.62 1.75 2.00 2.12 2.12 2.00 1.62 1.25

■Figure 7.14 The shoe profile for Example 7.1.

We have divided the profile into two parts and each part into 12 trapezoids of equal heights

of 1.0 in. Applying the trapezoidal rule, we have

Aha

1

2

y 0 y 1 y 2 pyn 2 yn 1 

1

2

ynb

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