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

Name: Problem 22, Chapter 10: University Calculus: Early Transcendentals
Uploaded: 2025-05-28T14:59:32-08:00
Duration: 1 min 38 s
Channel: Nick Johnson
Description: Find the lengths of the curves. The spiral r=e^θ / √(2), 0 ≤θ≤π

step 1 All right, so here we have that R of theta is equal to E to the theta over the square root of 2,

Find the lengths of the curves. The spiral r=e^θ / √(2), ...

Key Concepts

Arc Length in Polar Coordinates

This concept involves finding the length of a curve represented in polar coordinates. The formula for the arc length of a polar curve r(?) is L = ? from ? to ? ?[r(?)^2 + (dr/d?)^2] d?, which combines both the function itself and its rate of change. It is essential for converting polar expressions into a form that can be integrated over a given interval.

Differentiation of Polar Functions

Differentiation in the context of polar coordinates is used to compute the derivative dr/d?, which represents how the radius changes with respect to the angle. This derivative is a crucial part of the polar arc length formula because it captures the curve's rate of change and thus influences the overall length calculation.

Definite Integration Techniques

Definite integration is used to evaluate the arc length over a specified interval for the variable ?. The process involves integrating an expression that often includes both the function r(?) and its derivative dr/d?. Mastery of integration methods, including substitution and handling exponential functions, is vital to successfully determining the exact length of the curve.

Exponential Functions

Exponential functions, such as the one given by r(?) = e^(?)/?2, are key in modeling growth or decay processes in various mathematical contexts, including polar curves. Understanding their properties is important, as it affects both the differentiation and the subsequent integration steps needed to compute the arc length.

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