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# How long is a piece of string? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Problem

How can we find the length of a curve?

More precisely, if we have the graph of a function $y=f(x)$, how can we find the length of the graph between $x=a$ and $x=b$, as shown in this sketch?

And if we have a curve given parametrically as $(x(t), y(t))$, how can we find the length of the curve between $t=a$ and $t=b$, as shown in the following sketch?

If you would like some guidance, have a look at the Structured prompts section.

Once you have general techniques for answering the above questions, have a go at these problems:

1. Find the length of the curve described parametrically by $x=a\cos t$, $y=b\sin t$ from $t=0$ to $t=\theta$. Where have you met this result before?

2. Find the length of the curve $y=a\cosh\dfrac{x}{a}$ from $x=0$ to $x=x_0$. (This curve is called a catenary, and this result is closely related to some of the other resources at this station.)