274 



and 



PRACTICAL MATHEMATICS 



S = - W dz 



4 3 



- 79-36 



145. Let the whole arc of a curve be divided into a very large 

 number of small elementary arcs. 



FIG. 87, 



Let l lt 1 2 , 1 3 . . . be the lengths of these arcs, 



a?!, # 2 , # 3 . . . their distances from the axis OY, 

 /2 2/3 their distances from the axis OX (Fig. 87). 



and 



Let the whole area rotate about the axis OX and S ox be the 

 area of the surface of the resulting surface of revolution. 

 The elementary length l l will describe a surface of area 

 The elementary length 1 2 will describe a surface of area 

 The total surface will be the sum of all these elementary 

 surfaces. 



Hence S ox = 27r{Z 1 2/ 1 + l$. 2 + I 3 y s } 



where s is the whole length of the curve and y is the height of 

 the centroid of that length of curve above the axis OX. 

 If the whole area rotates about the axis OY, then 



S oy = 



