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

(dy)/(dx)=0.08x

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.

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