key term - Spiral of Archimedes

Definition

The spiral of Archimedes is a type of spiral curve that was first studied by the ancient Greek mathematician Archimedes. It is a plane curve that is generated by a point moving away from a fixed point at a constant rate, while the radius vector from the fixed point rotates at a constant angular velocity.

5 Must Know Facts For Your Next Test

1. The spiral of Archimedes is often used to model natural phenomena, such as the growth patterns of certain plants and shells.
2. The equation of the spiral of Archimedes in polar coordinates is $r = a + b\theta$, where $r$ is the distance from the origin, $\theta$ is the angle, and $a$ and $b$ are constants.
3. The spiral of Archimedes has a constant rate of change in the radial distance as the angle increases, resulting in a self-similar curve.
4. The area enclosed by one complete turn of the spiral of Archimedes is given by the formula $\frac{\pi b^2}{2}$.
5. The arc length of a portion of the spiral of Archimedes can be calculated using the formula $\int_{\theta_1}^{\theta_2} \sqrt{a^2 + b^2} \, d\theta$.

Review Questions

• Explain how the spiral of Archimedes is defined in polar coordinates and describe its key properties.
  • The spiral of Archimedes is a plane curve defined in polar coordinates by the equation $r = a + b\theta$, where $r$ is the distance from the origin and $\theta$ is the angle. This curve has a constant rate of change in the radial distance as the angle increases, resulting in a self-similar spiral shape. The spiral of Archimedes is often used to model natural phenomena, such as the growth patterns of certain plants and shells, due to its unique geometric properties.
• Derive the formula for the area enclosed by one complete turn of the spiral of Archimedes.
  • To derive the formula for the area enclosed by one complete turn of the spiral of Archimedes, we can use the formula for the area in polar coordinates: $A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta$. Substituting the equation of the spiral of Archimedes, $r = a + b\theta$, we get: $A = \frac{1}{2} \int_{0}^{2\pi} (a + b\theta)^2 \, d\theta = \frac{\pi b^2}{2}$.
• Describe how the arc length of a portion of the spiral of Archimedes can be calculated using integration in polar coordinates.
  • The arc length of a portion of the spiral of Archimedes can be calculated using the formula for arc length in polar coordinates: $s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + (\frac{dr}{d\theta})^2} \, d\theta$. Substituting the equation of the spiral of Archimedes, $r = a + b\theta$, we get: $s = \int_{\theta_1}^{\theta_2} \sqrt{a^2 + b^2} \, d\theta$. This integral represents the arc length of the portion of the spiral between the angles $\theta_1$ and $\theta_2$.

Related terms

Polar Coordinates:

A coordinate system that specifies the location of a point in a plane by a distance from a reference point and an angle from a reference direction.

Arc Length:

The distance along a curve between two points on the curve.

Area in Polar Coordinates: The area enclosed by a curve in polar coordinates, which can be calculated using integration.