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Arc Length

# Arc Length - Arc Length In this section we are going to...

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Arc Length In this section we are going to look at computing the arc length of a function. Because it’s easy enough to derive the formulas that we’ll use in this section we will derive one of them and leave the other to you to derive. We want to determine the length of the continuous function on the interval . Initially we’ll need to estimate the length of the curve. We’ll do this by dividing the interval up into n equal subintervals each of width and we’ll denote the point on the curve at each point by P i . We can then approximate the curve by a series of straight lines connecting the points. Here is a sketch of this situation for . Now denote the length of each of these line segments by and the length of the curve will then be approximately, and we can get the exact length by taking n larger and larger. In other words, the exact length will be,

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Now, let’s get a better grasp on the length of each of these line segments. First, on each segment let’s define . We can then compute directly the length of the line segments as follows. By the Mean Value Theorem we know that on the interval there is a point so that, Therefore, the length can now be written as,
The exact length of the curve is then, However, using the definition of the definite integral , this is nothing more than, A slightly more convenient notation (in my opinion anyway) is the following. In a similar fashion we can also derive a formula for on . This formula is,

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Again, the second form is probably a little more convenient. Note the difference in the derivative under the square root! Don’t get too confused. With one we differentiate with respect to
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Arc Length - Arc Length In this section we are going to...

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