Machinery's Handbook, 31st Edition
72 AREA OF SURFACE OF REVOLUTION Approximate Method for Finding the Area of a Surface of Revolution.— The illustra- tion below is an example of the approximate method based on Guldinus rule for finding the surface area of a symmetrical body. In the illustration, the dimensions in common fractions are the known dimensions; those in decimals are found by actual measurements on a figure drawn to scale.
The surface area is found as fol- lows: First, the entire form is sepa- rated into such areas as are cylin drical, conical, or spherical, since their surface areas can be found by exact formulas. In the illustra- tion, the three-dimensional portion marked in the plane by ABCD is a cylinder, the area of the surface of which can be easily found. The top area EF is simply a circular area and can thus be computed separately. The remainder of the surface gener- ated by rotating line AF about the axis GH is found by the approxi- mate method. From point A , equal distances are set off on line AF . In the illustration, each division indi- cated is 1 ⁄
G
” 1 2
0.03 ”
E
F
0.51 ”
1.62 ” 1.36 ” 0.99 ” 0.70 ” 0.55 ”
1.52 ” 1.15 ” 0.84 ” 0.61 ”
1 ”
1 ” 1 2
B
A
1 ” 5 8
T
C
D
H
8 inch long. From the central or middle point of each of these parts a line is drawn at right angles to the axis of rotation GH , the length of these lines or diameters (the length of each is given in decimals) is measured, all these lengths are added together and the sum is multiplied by the length of one division set off on line AF (in this case, 1 ⁄ 8 inch), and this product is multiplied by p to give the approximate area of the surface of revolution. In setting off divisions 1 ⁄ 8 inch long along line AF , the last division does not reach all the way to point F , but only to a point 0.03 inch below it. The part 0.03 inch high at the top of the cup can be considered as a cylinder of 1 ⁄ 2 -inch diameter and 0.03-inch height, the area of the cylindrical surface of which is easily computed. By adding the various surfaces together, the total surface of the cup is found as follows: Cylinder, 1 5 ⁄ 8 in. diameter, 0.41 in. height 2.093 in 2 Circle, 1 ⁄ 2 in. diameter 0.196 in 2 Cylinder, 1 ⁄ 2 in. diameter, 0.03 in. height 0.047 in 2 Irregular surface 3.868 in 2 Total 6.204 in 2 Area of Irregular Plane Figure.— One of the most useful and accurate methods for determining the approximate area of a plane figure or irregular outline is known as Simp- son’s rule . In applying Simpson’s rule to find an area, the work is done in four steps: 1) The area is divided into an even number N of parallel strips of equal width W ; for example, in the accompanying diagram, the area has been divided into 8 strips of equal width. 2) The sides of the strips are labeled V 0 , V 1 , V 2 , etc., up to V N . 3) These heights V 0 , V 1 , V 2 ,…, V N are measured. 4) The values V 0 , V 1 , etc. are substituted in the following formula to find the area A of the figure:
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