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Chapter 1: Problem 6

How would you determine the length of a curved line?

Short Answer

Expert verified
To determine the length of a curved line, derive the function that describes the curve, use the derivative to compute an integral using the formula \(L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx\), where \(L\) is the length of the curve, \(a\) and \(b\) are the start and end points of the curve, respectively, and \([f'(x)]^2\) is the square of the derivative of the function. The solution of this integral gives the length of the curve.

Step by step solution

01

Understand the concept

The idea behind finding the length of a curved line is to approximate that curve using many small straight segments, and then sum the length of these segments. The smaller the segments, the better the approximation becomes. In calculus, this is done using an integral.
02

Understand the formula for the length of a curve

The formula for finding the length of a curve given by a function \(y = f(x)\) from \(a\) to \(b\) is generally given as: \(L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx \) where \(f'(x)\) is the derivative of the function \(f(x)\).
03

Use the formula to compute the arc length

To find the length of a specific curved line, define the function that describes the curve, find its derivative, plug the derivative into the formula above, and compute the integral from the appropriate limits of integration (the starting and ending points of the interval along the x-axis over which you're trying to find the arc length). Assuming the integral can be calculated analytically, solve for the definite integral. If it can't be solved analytically, approximate the definite integral.
04

Interpret the result

The result that obtained from the integration is the length of the curve from point \(a\) to \(b\) on the function \(f(x)\). This is the measure of how long the curve is within the interval.

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