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.
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