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Excerpt from "Arc length of function graphs, introduction" article. Includes GROUP widget only.

Excerpt from "Arc length of function graphs, introduction"

When we put this into our integral, it unfolds to look a bit more familiar.

$\begin{aligned} \int \sqrt{(d x)^{2} + (d y)^{2}} & = \int \sqrt{(d x)^{2} + (2 x d x)^{2}} \\ = \int \sqrt{(1 + (2 x)^{2}) (d x)^{2}} \\ = \int \sqrt{1 + 4 x^{2}} d x \end{aligned}$ ‍

I've been purposefully lazy about placing bounds on this integral, but now that everything inside the integral is in terms of

x

, with no

d y

's mucking it up, it makes sense to define the bounds of integration in terms of the

x

value, which in this case is from

- 2

2

$\begin{array}{r} \int_{- 2}^{2} \sqrt{1 + 4 x^{2}} d x \end{array}$ ‍

This looks like something we can compute. Well, actually, in this case, it turns out to be quite a tricky integral, but in this day and age, we can just plug integrals into a computer if we need to. The point is that the idea of approximating our curve's length with little lines, which was at first vaguely written with loose notation, has now turned into a concrete, computable integral.

For now, rather than getting bogged down with the details of this integral (there's plenty of that coming in the next article), I want to highlight some points from this example.

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- Sarah S
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