Solids of revolution

1
Solids of revolution
When an area is rotated through 2?, a solid
object is formed. If a curve is rotated, a hollow
object is formed. Both are known as solids of
revolution.
Volume of revolution about the x-axis
The area is bounded by the curve y f(x) , the
x-axis , x a and x b, is rotated through 2?
radians (360?), about the x-axis. The volume of
the solid is given by
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Volume of revolution about the x-axis
Example
Find the volume of the solid formed when the area
bounded by y x3,the x-axis, x 2 and
x 3, rotated through 2? about the x-axis.
Solution
3
Volume of revolution about the x-axis
Example
Find the volume of the solid formed when the area
bounded by y (x 1)(x 3)and the
x-axis, rotated through 360? about the x-axis.
Solution
4
Volume of revolution about the x-axis
Example
Prove that the volume of a cone with base r and
height h is
Solution
5
Volume of revolution about the x-axis
Example
Prove that the volume of a sphere is
Solution
6
Volume of revolution about the y-axis
The area is bounded by the curve y f(x) , the
y-axis , y c and y d, is rotated through 2?
radians (360?), about the y-axis. The volume of
the solid is given by
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Volume of revolution about the y-axis
Example
Find the volume of the solid formed when the area
bounded by y ?x, the y-axis, y 1 and
y 2, rotated through 2? about the y-axis.
Solution
8
Volume of revolution about the y-axis
Example
Find the volume of the solid formed when the area
bounded by y x3, the y-axis, y 1 and
y 8, rotated through 360? about the y-axis.
Solution
9
Rotating regions between curves
If two curves, y f(x) and y g(x), intersect
at a and b, and f(x) gt g(x) in the
interval a ? x ? b, then the volume of the solid
of revolution formed by rotating the region
between the curves about the x-axis is
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Rotating regions between curves
Example
The area enclosed between the curve y 4 x2
and the line y 4 2x is rotated
through 360? about the x-axis. Find the volume of
the solid generated.
Solution
4 2x 4 x2 ? x 0 or x 2