This document discusses finding the centroid of solids of revolution. It explains that the centroid of a solid generated by revolving a plane area about an axis will lie on that axis. To find the coordinates of the centroid, one takes the moment of an elementary disc about the coordinate axes and sums these moments by treating the discs as an integral. Two examples are worked through to demonstrate finding the coordinates of the centroid. Five exercises are then posed asking to find the centroid coordinates for solids generated by revolving given bounded regions.
2. 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.
3. THE MOMENT OF A SOLID of volume V, generated by revolving a
plane area about a horizontal or vertical axis, with respect to the
plane through the origin and perpendicular to the axis may be found
as follows:
1.Sketch the region, showing a representative strip and the
approximating rectangle.
2.Form the product of the volume, disc or shell generated by
revolving the rectangle about the axis and the distance of the
centroid of the rectangle from the plane, and sum for all the
rectangles.
3.Assume the number of rectangles to be indefinitely increased and
apply the fundamental theorem.
4. 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.
)z,y,x(C
If Myz
is the moment of the solid to the y-z plane through the origin and
perpendicular to the x-axis, then
xvMyz ⋅=
xzMSimilarly if 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
xyM
0=xyM
yvMxz ⋅=
5. 1. Determine the centroid of the solid generated by revolving the area
bounded by the curve y = x2
, 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
.
.0=x
h= 9-y
y = x2
y=y
y = 9
( 3 , 9 )
( x , y )
),,( zyxC
•
)0,6,0(C
EXAMPLE
,
8. xy 42
=42 −= xy .4=x
2. Find the coordinates of the centroid of the solid generated by
revolving the area within and the line about
EXAMPLE
)0,
8
5
,4(Canswer
units.cuV
5
216π
=
π= 27xzM
9. EXERCISES:
Determine the coordinates of the centroids of the solids generated
by revolving:
1. the first quadrant region bounded by the curve y = 4 – x2
about
the y –axis.
2. the third quadrant region bounded by the curve y = x3
and y = x
about y – 1 = 0.
3. the region bounded by the curve y2
= 4x , the lines x = 0 and y =
4, about x + 1 = 0.
4. the region bounded by the curves y = x2
and y2
= – x about the
line y+1 = 0.
5. the region bounded by the curves y = 6x – x2
and y = x2
– 2x
about x – 4 = 0.
10. EXERCISES:
Determine the coordinates of the centroids of the solids generated
by revolving:
1. the first quadrant region bounded by the curve y = 4 – x2
about
the y –axis.
2. the third quadrant region bounded by the curve y = x3
and y = x
about y – 1 = 0.
3. the region bounded by the curve y2
= 4x , the lines x = 0 and y =
4, about x + 1 = 0.
4. the region bounded by the curves y = x2
and y2
= – x about the
line y+1 = 0.
5. the region bounded by the curves y = 6x – x2
and y = x2
– 2x
about x – 4 = 0.