Q:

# How to calculate the length of a curved line?

A:

In order to find the length of a curved line, or arc length, it is necessary to know the function of the line. Once this is determined, differentiate the function, plug the equation into the arc length formula, and then integrate the formula.

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Credit: pagedooley CC-BY 2.0

1. Differentiate the function

For example, if the function given is y = (x - 1)^(3 / 2), the derivative is (dy / dx) = (3 / 2) * (x - 1)^(1 / 2).

2. Plug the equation into the arc length formula

The arc length formula is L = |(a..b) sqrt(1 + (dy / dx)^2) dx. Assuming the length to be analyzed ranges from x = 1 to x = 5, the given function, when plugged into this formula and simplified, looks like this: L = |(1..5) ((9 / 4)x - (5 / 4))^(1 / 2) dx.

3. Integrate the formula

Integrating the formula determines the arc length: L = ((4 / 9) * (2 / 3) * ((9 / 4)x - (5 / 4))^(3 / 2)) | (1..5) = (1 / 27) * sqrt(40)^3 - (1 / 27) * 8 = 9.07. Thus, the length of the curved line is 9.07 units, whether the units are in inches, millimeters or miles.

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