We find the length of a curved segment.

1 Arc Length

(problem 1a) Find the length of the curve from to using the arc length formula.



Arc length = .

(problem 1b) Find the length of the curve from to using the distance formula.

The endpoints of the line segment are and


Arc length = .

(problem 2a) Find the length of the curve from to



Arc length =

(problem 2b) Find the length of the curve from to



Arc length =

(problem 3a) Find the length of the curve from to



Arc length =

(problem 3b) Find the length of the curve from to



Arc length =

(problem 3c) Find the length of the curve from to .


Arc length =

(problem 3d) Find the length of the curve from to .


Arc length =

2 Video Lesson

Here is a detailed, lecture style video on arc length:
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3 Theoretical Justifications

In this section, we derive the arc length formula. Suppose is a differentiable function on the open interval and continuous on the closed interval . Let and let for . Note that and . Next, to approximate the arc length, connect the adjacent points and with straight line segments for From the distance formula, the length of the segment is

Now, we apply the Mean Value Theorem to on the interval to conclude that for some number in the interval . Thus,

Returning to the arc length, , of on , we have Taking the limit as , we define the arc length as

2024-09-27 13:57:56