Centroids of Solids of Revolution: Applications and Exercises, Slides of Differential and Integral Calculus

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TOPIC

APPLICATIONS

CENTRIODS OF SOLIDS OF

REVOLUTION

Center of gravity of a solid of revolution The coordinates of the centre of gravity of a solid of revolution are obtained by taking the moment of an elementary disc about the coordinate axis and then summing over all such discs. Each sum is then approximately equal to the moment of the total volume taken as acting at the centre of gravity. Again, as the disc thickness approaches zero the sums become integrals: 2 2 and 0 b x a b x a xy dx x y y dx    

When the area is revolved about the x-axis, the centroid is on that axis. Which means that, for solids generated by revolving the plane area about an axis, its centroid is on that axis, thus, giving one coordinate. C (x,y,z ) If M yz is the moment of the solid to the y-z plane through the origin and perpendicular to the x-axis, then

M v x

yz

xz Similarly if M is the moment of the solid with respect to the x-z plane through the origin, and perpendicular to the y-axis, then is zero, considering that solids generated by revolving a plane area only about a horizontal or vertical axis, where which is the moment about an x-y plane, is always zero. ( ). z xy M Mxy  0

1. Determine the centroid of the solid generated by revolving the area bounded by the curve , y = 9, and x = 0, about the y-axis. Since the axis of revolution is the y-axis, the centroid of the solid is on that axis, giving . h= 9-y y = x^2 y= y y = 9 ( 3 , 9 ) ( x , y )

EXAMPLE

EXERCISES:

Determine the coordinates of the centroids of the solids generated by revolving:

1. the first quadrant region bounded by the curve about the y –axis.
2. the third quadrant region bounded by the curve and y = x about y – 1 = 0.
3. the region bounded by the curve , the lines x = 0 and y = 4, about x + 1 = 0.
4. the region bounded by the curves and about the line y+1 = 0.
5. the region bounded by the curves and about x – 4 = 0.