Find the arc length of the curves. r=θ, 0 ≤θ≤2 π

Name: Problem 43, Chapter 8: Calculus Single Variable
Uploaded: 2021-06-01T00:16:57-08:00
Duration: 1 min 52 s
Channel: Carson Merrill
Description: Find the arc length of the curves. r=θ, 0 ≤θ≤2 π

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Polar Coordinates

Polar coordinates use a radius (r) and an angle (?) to represent points in the plane, which simplifies the representation of curves that have a natural circular symmetry or spiral shape. This coordinate system is especially useful when dealing with curves defined by equations where the radius is given as a function of the angle.

Arc Length in Polar Coordinates

The arc length of a curve given in polar coordinates can be calculated using the formula L = ??(r² + (dr/d?)²) d?. This formula combines the radial and angular changes to compute the total distance along the curve, taking into account both the direct distance outwards and the change in the path's direction as the angle varies.

Differentiation with Respect to the Angle

Finding the derivative dr/d? is a crucial step in computing the arc length of a polar curve. This derivative reflects how the radial distance changes as the angle increases. It is used in the arc length formula to account for the rate at which the curve moves away from or toward the origin.

Transcript

Please give Ace some feedback