# 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