EXAMPLES XV 221 



(7) A body is composed of a cone with a hemispherical base, 

 the radii of the base of the cone and the hemisphere each being 

 5 inches. Find the height of the cone so that the centroid of 

 the body shall lie in the surface common to the cone and hemi- 

 sphere. 



(8) Find the co-ordinates of the centroid of the section (Fig. 73, 

 No. 1) with reference to the axes OX and OY. 



(9) Find the height of the centroid of the section (Fig. 73, 

 No. 2) above the base AB. 



(10) Find the perpendicular distance of the centroid of the 

 section (Fig. 73, No. 4) from the side AB. 



(11) Find the co-ordinates of the centroid of the section (Fig. 73, 

 No. 6) with reference to the axes OX and OY. (The full depth is 6".) 



(12) Find the perpendicular distance of the centroid of the 

 section (Fig. 73, No. 7) from the side AB. 



(13) Find the distance from the centre, of the centroid of a 

 quadrant of a circle of radius 4 inches. 



(14) ABCD is a square, 8 inches side. From the corner D a 

 quadrant of a circle, 4 inches radius, is cut away. Find the 

 co-ordinates of the centroid of the remainder, with reference to 

 the sides AB and BC as axes. 



(15) Find the co-ordinates of the centroid of the area bounded 

 by the curve y = 3x 2 , the axis of x, and the ordinates at x = 

 and x = 3. 



(16) Find the co-ordinates of the centroid of the area bounded 

 by the curve y = x 2 Qx + 18 and the axes of reference. 



(17) Find the co-ordinates of the centroid of the area bounded 

 by the curve y = x z 9x + 18 and the axis of x. 



(18) Find the co-ordinates of the centroid of the area enclosed 

 by the two curves y 2 = 8x and x 2 = 8y. 



(19) Find the co-ordinates of the centroid of the area bounded 

 by the curve y = 5 V^c, the axis of x, and the ordinates at x = 2 

 and x = 4. 



(20) Find the co-ordinates of the centroid of the quadrant of 



x 2 ij 2 

 an ellipse, the equation of the ellipse being + ^ 1. 



(21) The curve y = ax n passes through the points (2, 5-37) and 

 (5, 28-62). Find a and n. Find the co-ordinates of the centroid 

 of the area bounded by the curve, the axis of x, and the ordinates 

 at x = 2 and x = 5. 



(22) Find the first two points at which the curve y = e* sin x 

 crosses the axis of x. Find the height above the axis of x, of the 

 centroid of the area bounded by the curve and the axis of x 

 between these points. 



(23) The curve y = x 2 + 5 is cut by the line y = 4# + 5. Find 

 the co-ordinates of the centroid of the area enclosed by the curve 

 and the line. 



