Definition:Solid of Revolution
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Definition
Let $F$ be a plane figure.
Let $L$ be a straight line in the same plane as $F$.
Let $F$ be rotated one full turn about $L$ in $3$-dimensional space.
The solid of revolution generated by $F$ around $L$ is the solid figure whose boundaries are the paths taken by the sides of $F$ as they go round $L$.
The usual scenario is that:
- $(1): \quad$ one of the sides of $F$ is a straight line coincident with $L$
- $(2): \quad$ $L$ itself is aligned with one of the coordinate axes, usually the $x$-axis in a cartesian plane
- $(3): \quad$ Two other sides of $F$ are also straight lines, perpendicular to $L$
- $(4): \quad$ The remaining side (or sides) of $F$ are curved and described by means of a real function.
The above diagram shows a typical solid of revolution.
The plane figure $ABCD$ has been rotated $2 \pi$ radians around the $x$-axis.
$FECD$ illustrates the disposition of $ABCD$ part way round.
Axis of Revolution
The straight line is known as the axis of revolution.
Also known as
A solid of revolution is also known as a solid (or volume) of rotation.
Also see
- Results about solids of revolution can be found here.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Area, Volume and Centre of Gravity: Volume of Rotation about the $x$-axis