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36. Length of Plane Curves

Dated: 30-10-2024

The Arc Length Problem

The continuous functions1 are also called smooth functions and their graphs are called smooth curves.
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We can divide the curve into small enough intervals2 such that the distances between the points on the curve appear to be within a straight line.3
This way, we can cover the whole curve and accurately measure the arc length.

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If we want to know the length of \(\overline{P_0 P_1}\), we can use pythagorus theorem.

\[m\overline{P_0 P_1} = \sqrt{\Delta x^2 + \Delta y^2}\]
\[\because \Delta x = x_1 - x_0\]
\[\because \Delta y = f(x_1) - f(x_0)\]
\[m\overline{P_0 P_1} = \sqrt{(x_1 - x_0)^2 + (f(x_1) - f(x_0))^2}\]

Let's generalize this equation.

\[L_i = m \overline{P_{i - 1} P_i}\]
\[L_i = \sqrt{(x_{i} - x_{i - 1})^2 + (f(x_i) - f(x_{i - 1}))^2}\]

According to mean value theorem4

\[\frac{f(x_i)-f(x_{i-1})}{x_i-x_{i-1}}=f'(x_i^*)\]
\[\Rightarrow f(x_i)-f(x_{i-1})=f'(x_i^*)(x_i-x_{i-1})=f'(x_i^*)\Delta x_i\]

Plugging these into our equation, we get

\[L_i=\sqrt{(\Delta x_i)^2+(f'(x_i^*))^2(\Delta x_i)^2}\]
\[L_i=\sqrt{(\Delta x_i)^2(1 + (f'(x_i^*))^2)}\]
\[L_i=\sqrt{1+(f'(x_i^*))^2}\Delta x_i\]

Therefore, for the whole polygonal path, we have

\[\sum_{i=1}^{n}L_{i}=\sum_{i=1}^{n}\sqrt{1+(f^{\prime}(x_{i}^{*}))^{2}}\Delta x_{i}\]

And we know that our approximation gets better as the distance between the intervals2 decreases, therefore,

\[L=\lim_{\max(\Delta x_i\to 0)}\sum_{i=1}^{n}\sqrt{1+(f^{\prime}(x_{i}^{*}))^{2}}\Delta x_{i}\]

Of course, we can re-write it in integral5 form.

\[L=\int_{a}^{b}\sqrt{1+(f^{\prime}(x))^{2}}dx\]

References

Read more about notations and symbols.

  1. Read more about continuity. ↩

  2. Read more about intervals. ↩↩

  3. Read more about lines. ↩

  4. Read more about mean value theorem. ↩

  5. Read more about integrals. ↩