R1(θ) = r2(θ)

Definition

The equation r1(θ) = r2(θ) represents a condition where the radial functions of two polar curves are equal at the same angle θ. This equality of the radial functions is an important concept in the study of area and arc length in polar coordinates.

5 Must Know Facts For Your Next Test

1. The condition r1(θ) = r2(θ) indicates that the two polar curves have the same radial distance from the origin at the same angle θ, meaning they intersect at that point.
2. This equality of radial functions is crucial for calculating the area bounded by two polar curves, as well as the arc length of a single polar curve.
3. When r1(θ) = r2(θ), the area between the two polar curves is zero, as they occupy the same space.
4. The arc length of a polar curve is calculated using the formula $\int_{\theta_1}^{\theta_2} r(\theta) \, d\theta$, where r(θ) is the radial function.
5. If r1(θ) = r2(θ), the arc length formula simplifies to $\int_{\theta_1}^{\theta_2} r(\theta) \, d\theta$ for either curve, as they have the same radial function.

Review Questions

• Explain the significance of the condition r1(θ) = r2(θ) in the context of calculating the area bounded by two polar curves.
  • When the radial functions of two polar curves are equal, r1(θ) = r2(θ), the area between the curves is zero. This is because the curves occupy the same space and do not enclose any region. The area formula for the region between two polar curves, $\int_{\theta_1}^{\theta_2} \frac{1}{2} [r_2^2(\theta) - r_1^2(\theta)] \, d\theta$, simplifies to zero when r1(θ) = r2(θ). This equality of radial functions is a crucial condition to consider when determining the area bounded by polar curves.
• Describe how the condition r1(θ) = r2(θ) affects the calculation of arc length for a single polar curve.
  • The arc length of a polar curve is calculated using the formula $\int_{\theta_1}^{\theta_2} r(\theta) \, d\theta$, where r(θ) is the radial function. When r1(θ) = r2(θ), meaning the two polar curves have the same radial function, the arc length formula simplifies to $\int_{\theta_1}^{\theta_2} r(\theta) \, d\theta$ for either curve. This is because the curves occupy the same space, and the arc length calculation can be performed using the shared radial function r(θ).
• Analyze the relationship between the condition r1(θ) = r2(θ) and the concept of polar curves intersecting at a point.
  • The equality of radial functions, r1(θ) = r2(θ), indicates that the two polar curves intersect at the angle θ. This is because the curves have the same distance from the origin at that specific angle, meaning they occupy the same point in the polar coordinate system. The intersection of the curves is a crucial consideration when studying the area and arc length of polar curves, as the condition r1(θ) = r2(θ) simplifies the calculations and reveals important properties about the curves' relationship.

Related terms

Polar Coordinates: A coordinate system that specifies the location of a point on a plane using a distance from a fixed origin (the radius or radial coordinate r) and an angle from a fixed reference direction (the angular coordinate θ).

Radial Function: In polar coordinates, the radial function r(θ) defines the distance from the origin to a point on the curve as a function of the angle θ.

Polar Curve: A curve in the polar coordinate system, where the position of a point on the curve is determined by its radial distance r and the angle θ.