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Mathematical Expression Editor
We find the length of a curved segment.
1 Arc Length
Arc Length The length of the graph of the differentiable function, , from to , is given
by
example 1 Find the length of the curve from to . Since the graph of this function is a line, calculus is not required to find the length of
the indicated segment. Instead, we can use the distance formula, with the points
and to get We will also use the formula from the above theorem as a verification
that the theorem is correct. Since , we have and the length of the segment is as
expected.
(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 = .
example 2 Find the length of the graph of from to . First, and so . Then the arc length is To compute this integral, we will use a
-substitution with and . Thus, the anti-derivative is,
Returning to our computation of 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 =
example 3 Find the length of the graph of from to . In this example, computing a simplified form of the integrand is laborious, so
we will proceed slowly. First, , so and Next, we add one: and the key
to this problem is that this expression is a perfect square: Now, we can
compute the square root: Finally, we can integrate to get the arc length: \begin{align*} L &= \int _1^2 \sqrt{1+f'(x)^2} \; dx \\ &= \int _1^2 \left (\frac{x^2}{4} + \frac{1}{x^2} \right ) \; dx \\ &= \int _1^2 \left (\frac{x^2}{4} + x^{-2} \right ) \; dx \\ &= \left (\frac{x^3}{12} + \frac{x^{-1}}{-1} \right ) \bigg |_1^2 \\ &= \left (\frac{x^3}{12} - \frac{1}{x} \right ) \bigg |_1^2 \\ &= \left (\frac 23 - \frac 12\right ) - \left (\frac{1}{12} - 1\right ) \\ &= \frac{13}{12}. \end{align*}
(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