An **axis of rotation** (also called an *axis of revolution*) is a line around which an object rotates. In calculus and physics, that line is usually imaginary. The **radius of rotation** is the length from the axis of rotation to the outer edge of the object being rotated.

## Example of Axis of Rotation

A simple example is **one axle **or hinge that allows rotation, but not translation (movement). The following image shows a two-dimensional shape (a half bell) rotating around a single, vertical axis of rotation. If the shape travels 360 degrees, the result is a three-dimensional bell:

The final object (in this case, a bell) is called an **object of revolution.**

## Radius of Rotation Example

The radius of rotation is a line segment that extends from the rotational axis to a point of interest on the outer edge.

If you’re using the disc method, the radius of rotation is perpendicular (at right angles) to the axis of rotation.

## Real Life Axis of Rotation Examples

## 1. Aviation

In **aviation**, “axis of rotation” refers to one of three axes, about which an airplane pitches, rolls, or yaws.

The three axes (shown in the above image with blue arrows) are:

**Lateral (pitch)**, controlled by the elevator,**Longitudinal (roll)**, controlled by ailerons.**Vertical (yaw)**, controlled by the rudder.

## 2. Earth’s Axis of Rotation

The **Earth** is tilted at an angle of 23.5° with respect to the Sun. The axis of rotation affects when the seasons happen;

- When the tilt is towards the Sun, it’s summer in the northern hemisphere and winter in the southern hemisphere.
- When the tilt is away from the Sun, it’s winter in the northern hemisphere and summer in the southern hemisphere.

The Earth’s tilt is much like a spinning top; It wobble’s as it orbits the Sun, creating a *wobble effect.*

## Finding the Volume of a Solid of Revolution

The **disc method** or **washer method** are used to find the volume of objects of revolution in calculus. The disc method is used for solid objects, while the washer method is a modified disc method for objects with holes. More specifically:

- Use the
**disc method**if:- The axis of rotation is also the boundary (edge) of the two-dimensional object you’re rotating,
*and* - The cross sections (thin slices of the resulting solid) are taken perpendicular to the axis of rotation.

- The axis of rotation is also the boundary (edge) of the two-dimensional object you’re rotating,
- Use the
**washer method**if:- The axis of rotation isn’t a boundary of the two-dimensional object, and
- The cross sections are taken perpendicular to the axis of rotation.

## References

Smithsonian National Air and Space Museum. How Things Fly. Retrieved October 10, 2019 from: https://howthingsfly.si.edu/flight-dynamics/roll-pitch-and-yaw

Stewart, J. (2009). Calculus: Concepts and Contexts. Retrieved October 8, 2019 from: