Under construction Resources under construction may not yet have been fully reviewed.

# 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

## Structured prompts

1. What is the length of this line (not drawn accurately)?

2. What about the length of this approximation to $y=\frac{1}{2}x^2$ from $x=0$ to $x=3$ (again not drawn accurately)?

3. How could we get a better approximation to the length of $y=\frac{1}{2}x^2$ from $x=0$ to $x=3$?

4. How can you extend your answer to work out an expression for the exact length of $y=\frac{1}{2}x^2$ from $x=0$ to $x=3$?

(Feel free to leave the expression in an unevaluated form if you do not know how to evaluate it.)

5. Extend this idea to answer the first main problem:

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?

6. Can you now extend your ideas to answer the second main problem?

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?

With these answered, you should be able to tackle the two examples on the main problem page.