- #1

MxwllsPersuasns

- 101

- 0

## Homework Statement

The problem statement is in the attached picture file and this thread will focus on question 7

## Homework Equations

The length of a curve formula given in the problem statement

Take a polygon in R^n as an n-tuple of vectors (a

_{0},...,a

_{k}) where we imagine the vectors, a

_{i}, as the vertices of the polygon and two successive vertices (a

_{i}and a

_{i+1}) be connected by the corresponding segment of straight line.

- Thus, the length of a polygon is the sum of the lengths of the line segments: Σ

^{k-1}

_{i=0}{ absval(a

_{i+1}-a

_{i})}

## The Attempt at a Solution

Basically this question is asking to show that the length of a polygonal curve and a smooth curve are equivalent once we consider an infinite amount of partitions of the interval [a,b]. I'm not exactly sure how to write this mathematically but I would approach the problem as follows:

I would show using the length of the curve formula that the total length of the curve was the sum of the integrals of each equidistant partition i.e., take an integral from (a=)t

_{0}to t

_{1}of γ'(t) then another integral from t

_{1}to t

_{2}all the way to the last integral of t

_{n-1}to t

_{n}(=b) and then try to show that the line segment joining the vertices γ(t

_{0}) and γ(t

_{1}), say, had the same length as the integral of the curve from t

_{0}to t

_{1}and then basically just say as you get closer to ∞ line segments the two lengths converge.

Is this on the right track in terms of concept? Anyone who can help me with the approach or helping to translate it into a mathematical argument would be very much appreciated!