THE CENTROID 255 



Taking the second strip and treating it as a rectangle, 

 Area of the strip = by 2 



8b 

 Distance of the centroid of the strip from OY = 



Moment of the strip about OY = ~b 2 y 2 . 



The whole area will be obtained by taking the sum of all these 

 strips. 



A= %!+2/2+2/3+ i/J 



The moment of the whole area will be obtained by taking the 

 sum of the moments of all these strips. 



M _!&2 + ? 6 2 v +5fc2, , 2n ~ 1 fr 



{2/i + 32/2 + 52/3 + ... (2n - !)*/} 



If x is the perpendicular distance of the centroid from the 

 axis OY, 



*- 



2 2/i + 2/2 + 2/3 + y n 



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

 surface of revolution, 



V oy = 27rAJ 

 = 2rc M OY 



Let the side OC be divided into n equal parts and the figure 

 divided into n strips of equal breadth by lines drawn through 

 the points of division, parallel to the axis OX. Let a be the 

 breadth of each strip, and x lt x z , x 3 . . . x n the mid-ordinates 

 of the strips. 



Then A = a(x^ + x 2 + x z + . . . x n ) 



M ox = {i 



. . . (2n - l)x n } 



If y is the perpendicular distance of the centroid from the 

 axis OX, 



y 



