Estimate: Looking at the graph (which has equal scaling along the x- and y-axes), we can see the final answer should be a little more than 10 m, somewhere between 10 m and 11 m.

Now to find the exact length:

`text[length]=r=int_a^b sqrt[1+((dy)/(dx))^2]\ dx`

In this example,

y = 0.04x2, so


The lower limit is x = −5 and the upper limit is x = 5. Substituting these into our formula gives:

`r=int_-5^5 sqrt(1+(0.08x)^2)\ dx = 10.261`

[Note: I used a computer algebra system to find the above integral. Many arc length problems lead to impossible integrals. (This example does have a solution, but it is not straightforward.) Often the only way to solve arc length problems is to do them numerically, or using a computer. You can see the answer in Wolfram|Alpha.]

So the length of the steel supporting band should be 10.26 m.

This is consistent with our earlier estimate.

Easy to understand math videos: