210 



PRACTICAL MATHEMATICS 



If the area rotates about the axis of y, another surface of 

 revolution is described, which can be taken as the sum of a large 

 number of elementary rings whose volumes will be 2Tta l x 1 , 2na z x z , 

 2na 3 x 3> and so on. 



Then V OY = 2na 1 x 1 + 2iia 2 x 2 + 27ra 3 # 3 + . . . 

 = 2n{a 1 x 1 



These results can be expressed in this way : " That if an area 

 rotates about an axis in its plane, the volume of the surface of re- 

 volution generated will be given by the area multiplied by the cir- 

 cumferential distance travelled by the centroid in one revolution." 



X 



FIG. 52, 



Also if x and y are the co-ordinates of the centroid of a given 

 area 



V, 



OY 



and 



These relations enable us to find the position of the centroid, 

 providing we know the area and the volumes of the surfaces of 

 revolution described as that area rotates about the two axes of 

 reference. 



117. Example. The curve y = ax n passes through the points 

 (2, 5) and (4, 11). Find the values of the constants a and n. 

 Taking A as the area bounded by the ordinates at as = 2 and 

 x = 4, and B as the area bounded by the abscissae corresponding 

 to the ordinates at x = 2 and x = 4 ; find the positions of the 

 centroids of the areas A and B. 

 Now 5 = a2 n 



11 = 4 n 

 2-2 = 2 n , and n = 1-137 



