A manufacturer of corrugated metal roofing wants to produce panels that are u = 48 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave with equation y = sin(πx/12).

Question

Find the width w of a flat metal sheet that is needed to make a 48-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

Answer 1

I assume that the center of the wavy sheet follows the sine curve. Otherwise you have to worry about two curves, for both top and bottom.

Now, he period of sin(πx/12) is (2π)/(π/12)=24

∫[0,48] √[1+(√π/12 cos(π/12 x))^2] dx

Still stuck with that elliptic integral, I see. There are lots of online integration sites which can evaluate it for you. wolframalpha gets 48.2607

http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0,48%5D+%E2%88%9A%5B1%2B(%E2%88%9A%CF%80%2F12+cos(%CF%80%2F12+x))%5E2%5D+dx

Answer 2

Thank you Steve. I can't see to find anyway to do this question on paper. My research show that I need an integrated computer to solve this hard one but I don't know what it is.

Answer 3

Nor am I. I suppose if you can add special functions to your repertoire, such as Gamma, Beta, Erf, etc. Then you can just use them in your analysis. I mean, we use stuff like sin(x) all the time, but it's a pretty nasty animal that you have to look up in a table to really understand. Or, for that matter, ln(x), which is defined as ln(x)=∫[1,x] 1/x dx.

So, good luck along those lines, but you won't be able to do this one on paper using elementary functions.