Consider the length of the graph of f(x)=5 / x from (1,5) to...

Consider the length of the graph of $f(x)=5 / x$ from $(1,5)$ to $(5,1) .$ (a) Approximate the length of the curve by finding the distance between its two endpoints, as shown in the first figure. (b) Approximate the length of the curve by finding the sum of the lengths of four line segments, as shown in the second figure. (c) Describe how you could continue this process to obtain a more accurate approximation of the length of the curve.

Consider the length of the graph of $f(x)=5 / x$ from $(1,5)$ to $(5,1) .$
(a) Approximate the length of the curve by finding the distance between its two endpoints, as shown in the
first figure.
(b) Approximate the length of the curve by finding the sum of the lengths of four line segments, as shown in the second figure.
(c) Describe how you could continue this process to obtain a more accurate approximation of the length of the curve.

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Key Concepts

Limit Process for Arc Length

The process of refining the polygonal approximation by increasing the number of segments leads to the concept of a limit, where the sum of the segment lengths approaches the actual arc length of the curve. In calculus, this idea is formalized by taking the limit as the number of segments goes to infinity, resulting in the integral formulation for arc length. This approach demonstrates how a continuous curve can be approximated by discrete segments to achieve an increasingly accurate measurement.

Distance Formula

The distance formula calculates the straight-line (Euclidean) distance between any two points in the plane using the coordinates of the points. In the context of curve length approximation, it is used to find the distance between the endpoints of the curve, which gives the chord length. This is the simplest form of approximation that ignores any curvature between the endpoints.

Polygonal Approximation

Polygonal approximation involves approximating the length of a curve by dividing it into a finite number of straight-line segments. Each segment connects two successive points on the curve. The sum of the lengths of these segments, calculated using the distance formula, provides a better approximation of the actual curve length than simply measuring the chord between the endpoints.

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