# Definition:Surface 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 **surface of revolution generated by $F$ around $L$** is the surface of the solid of revolution generated by $F$ around $L$.

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 **surface 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.