Find the exact length of the polar curve. r=θ^2, 0 ≦θ⩽2 π...

Name: Problem 35, Chapter 9: Essential Calculus Early Transcendentals
Uploaded: 2020-06-07T21:37:28-08:00
Duration: 5 min 43 s
Channel: Chris Trentman
Description: Find the exact length of the polar curve. r=θ^2, 0 ≦θ⩽2 π

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Key Concepts

Polar Coordinates

Polar coordinates describe points in the plane using a radius and an angle. This system is especially useful for curves that are naturally expressed in terms of a distance from a fixed point and a direction, facilitating the analysis of curves that might be cumbersome in Cartesian coordinates.

Arc Length in Polar Coordinates

The length of a curve expressed in polar form is determined by the integral L = ??(r² + (dr/d?)²)d? over the given interval. This formula accounts for both the radius and the rate of change of the radius with respect to the angle, encapsulating the total change along the curve.

Differentiation

Differentiation, specifically finding dr/d?, is crucial in applying the arc length formula, as the derivative describes how the radius changes with respect to the angle. This derivative is squared and added to the square of the radius function, forming a key part of the integrand.

Integration

Integration is used to sum up the small arc lengths along the curve to obtain the total length. Evaluating the integral involves techniques to handle the square root of the sum of squares, and may require methods such as substitution or algebraic manipulation to achieve an exact result.

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