What is the Square Footage of the Barn Roof?

[icesalmon]

Homework Statement

A barn is 100 feet long and 40 feet wide. A cross section of the roof is the inverted catenary y = 31 - 10(e^x/20+e^-x/20). Find the number of square feet of roofing on the barn

Homework Equations

Arc Length Integral
Hyperbolic trigonometric identities

The Attempt at a Solution

I start with my bounds [-20,20] and by rewriting y = f(x) as 31 - 20cosh(x/20) and continue on with y' as 0 - 20sinh(x/20)*(1/20) -> -sinh(x/20). y'² = sinh²(x/20). The Arc Length formula requires me to take the square root of 1 + y'². But I feel I should multiply my overall arc length by 100 to give me the area in square feet of the roof. So I have as my integrand 100(1+sinh²(x/20))^1/2. Using the identity sinh²(x) = (cosh(2x) - 1)/2 I have for my integrand 100(1+(cosh(x/10)-1)/2)^)1/2) which boils down to 100((cosh(x/10) +1)/2)^1/2 and I can use the identity (cosh(2x)+1)/2 = sinh²(x) to move it into 100sinh(x/20). Letting u = x/20 -> 20du = dx. After integration I have 2,000(sinh(x/20) evaluated at x=20 and x= -20 to give me 2,000(sinh(1)-sinh(-1))ft². Hopefully I did everything right, I don't have an answer in my book.

 

[TSny]

That looks good to me. Using the fact that sinh(x) is an odd function, you should be able to simplify your final answer.

 

[icesalmon]

of course, thank you.
I believe I just double the area, 4,000(sinh(1)ft² should be the value if that's true.