256 PRACTICAL MATHEMATICS 



If the area is made to rotate about the axis OX describing a 

 surface of revolution, 



= 27rM ox 



= na z {x 1 + 3x z + 5x 3 + . . . (2n - l)x n } 



138. The Moment of Inertia. Considering the figure (Fig. 80) 

 to be divided into vertical strips, each of breadth b, and y lt y 2 , 

 ?/ 3 . . . y n being the mid-ordinates of the strips. 



Taking each strip as a rectangle, 



Moment of inertia of the first strip about the axis OY = Ij, 



and I = + 



For the second strip I 2 = TH b 3 y z + by z x ( ^ 



. (2n-l)6\2 



For the last strip I n = b s y n + by n x 



The moment of inertia of the whole figure about the axis OY 

 will be obtained by taking the sum of the moments of inertia 

 of all these strips. 



b 3 b 3 



IOY= &i + 2/2 + 2/3+ Vn] + - {*/i + 9y z 



= 12 A+ 4 { ^ + 9 2/2+25?/3+ . . . (2n-l) 2 ?/ n } 



If the figure is divided into horizontal strips each of breadth 

 a, and x lt x z , x 3 . . . x n are the mid-ordinates of the strips, 



Then 



= A + 



