How long is a piece of string?

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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.