Find the lengths of the curves. The spiral r=θ^2, 0 ≤θ≤√(5)

Name: Problem 21, Chapter 10: University Calculus: Early Transcendentals
Uploaded: 2020-04-19T19:39:46-08:00
Duration: 7 min 21 s
Channel: Carlos Pinilla
Description: Find the lengths of the curves. The spiral r=θ^2, 0 ≤θ≤√(5)

step 1 So we would like to compute the length of the spiral r equals to theta squared. R equals theta s

Find the lengths of the curves. The spiral r=θ^2, 0...

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a fixed reference point (the origin) and an angle from a fixed direction. This concept is essential in studying curves that are naturally or more conveniently described by a radius and an angle rather than by Cartesian coordinates.

Arc Length in Polar Coordinates

The arc length of a curve defined in polar coordinates is calculated using an integral that incorporates both the radius function and its derivative with respect to the angle. This formula, L = ? ?(r² + (dr/d?)²) d?, is derived by applying the Pythagorean theorem to infinitesimal elements of the curve, adapting the standard arc length formula to the polar coordinate system.

Differentiation

Differentiation is a fundamental concept in calculus used to find the rate at which a function changes. In the context of finding arc lengths, differentiation is used to obtain the derivative of the radius function with respect to the angle, which is a critical component in the arc length formula for curves expressed in polar coordinates.

Integration

Integration is the process of summing up infinitely many small quantities to find a total accumulation, such as area or length. In calculating the arc length of a curve, integration is employed to sum up the contributions of each infinitesimal segment of the curve, resulting in the total length over the interval of interest.

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