R(θ)

Definition

r(θ) is the polar coordinate function that represents the distance from the origin to a point on a curve as a function of the angle θ. It is a fundamental concept in the study of polar coordinates and their applications in areas such as arc length and area calculations.

5 Must Know Facts For Your Next Test

1. The polar coordinate function r(θ) represents the distance from the origin to a point on a curve as a function of the angle θ.
2. The value of r(θ) can vary depending on the angle θ, allowing for the creation of different shapes and curves in the polar coordinate system.
3. The polar coordinate function r(θ) is essential for calculating the arc length and area of regions in polar coordinates.
4. The graph of r(θ) is called a polar curve, and its shape is determined by the specific function used to define r(θ).
5. Understanding the properties and behavior of r(θ) is crucial for solving problems involving polar coordinates, such as finding the area bounded by a polar curve or the arc length of a segment of a polar curve.

Review Questions

• Explain the relationship between the polar coordinate function r(θ) and the shape of a polar curve.
  • The polar coordinate function r(θ) directly determines the shape of a polar curve. The value of r(θ) represents the distance from the origin to a point on the curve, and as the angle θ varies, the value of r(θ) changes, resulting in the formation of different shapes and curves in the polar coordinate system. The specific function used to define r(θ) will dictate the characteristics and appearance of the resulting polar curve.
• Describe how the polar coordinate function r(θ) is used to calculate the arc length of a segment of a polar curve.
  • To calculate the arc length of a segment of a polar curve, the polar coordinate function r(θ) is used in conjunction with the formula for arc length in polar coordinates. The formula for arc length is given by: $$\int_{\theta_1}^{\theta_2} r(\theta) \, d\theta$$, where $\theta_1$ and $\theta_2$ represent the starting and ending angles of the arc segment. By substituting the specific function for r(θ) into this integral, the arc length of the desired segment can be calculated.
• Explain how the properties of the polar coordinate function r(θ) can be used to determine the area bounded by a polar curve.
  • The polar coordinate function r(θ) is also essential for calculating the area bounded by a polar curve. The formula for the area of a region bounded by a polar curve is given by: $$\frac{1}{2} \int_{\theta_1}^{\theta_2} r^2(\theta) \, d\theta$$, where $\theta_1$ and $\theta_2$ represent the starting and ending angles of the region. By understanding the properties and behavior of the function r(θ), including its dependence on the angle θ, one can substitute the specific form of r(θ) into this integral to determine the area of the desired region in the polar coordinate system.

Related terms

Polar Coordinates: A coordinate system that specifies the location of a point in a plane by a distance from a reference point (the origin) and an angle from a reference direction (the polar axis).

Polar Axis: The reference direction from which the angle θ is measured in a polar coordinate system.

Polar Curve: A curve that is defined by the polar coordinate function r(θ), which describes the distance from the origin to a point on the curve as a function of the angle θ.