220 PRACTICAL MATHEMATICS 



Taking a vertical strip of breadth S, 



Moment of strip = - y 2 # 

 2 



1 f' 



Total moment = - I u 2 dx 



1 f T 



= - I (1 cos 2x) dx 



4 Jo 



i 



= -{TT - (sin 27C sin 0)} 



Height of centroid = 



7T 



4 



total moment 

 area 



H 



8 



EXAMPLES XV 



(1) ABCD is a square, 5 inches side ; E is a point taken in 

 AD such that AE = 3 inches ; F is a point taken in CD such 

 that CF = 1 inch. Find the co-ordinates, with reference to the 

 sides AB and BC as axes, of the centroid of the figure ABCFE. 



(2) Prove that the centroid of the trapezium, whose parallel 

 sides are a and b respectively and whose height is h, is situated 



above the side b at a height of - r . 



3 a+ b 



(3) Find the height above the base, of a semicircular area of 

 radius a. 



(4) Show that the centroid of a hemisphere of radius a is situ- 

 ated on its axis at a distance - from the base. 



8 



(5) Show that the centroid of a cone is situated on its axis at 



a distance - from the base, where h is the height of the cone. 



4 



(6) A body is composed of a cylinder with a hemispherical 

 base, the radii of the cylinder and hemisphere each being 5 inches. 

 Find the height of the cylinder so that the centroid of the body shall 

 lie in the surface common to the cylinder and hemisphere. 



