A closed barrel has as its curved surface the surface obtained by rotating about the x-axis the part of the curve y = a[ 2 − cosh(x/a) ] lying in the range −b ≤ x ≤ b, where b

Question

A closed barrel has as its curved surface the surface obtained by rotating about the x-axis the part of the curve

y = a[ 2 − cosh(x/a) ]

lying in the range −b ≤ x ≤ b, where [latex]b

A = [latex]\pi[/latex]a[ 9a − 8a exp(−b/a) + a exp(−2b/a) − 2b ].

Accepted Answer

If s is the length of the curve defining the surface (measured from x = 0) then ds² = dx² + dy² and consequently [latex]d s / d x=\left(1+y^{\prime 2} ight)^{1 /2}[/latex].

For this particular surface,

[latex]y=a\left(2-\cosh \frac{x}{a} ight)[/latex]

and [latex]\frac{d y}{d x}=-\sinh \frac{x}{a}.[/latex]

It follows that

[latex]\begin{aligned}\frac{d s}{d x} & =\left[1+\left(\frac{d y}{d x} ight)^2 ight]^{1 / 2} \& =\left(1+\sinh ^2 \frac{x}{a} ight)^{1 / 2} \& =\cosh \frac{x}{a}.\end{aligned}[/latex]

The curved surface area, [latex]A_1[/latex], is given by

[latex]\begin{aligned}A_1 & =2 \int_0^b 2 \pi y d s \& =2 \int_0^b 2 \pi y \frac{d s}{d x} d x \& =4 \pi a \int_0^b\left(2 \cosh \frac{x}{a}-\cosh ^2 \frac{x}{a} ight) d x, ext { use } \cosh ^2 z=\frac{1}{2}(\cosh 2 z+1), \& =4 \pi a \int_0^b\left(2 \cosh \frac{x}{a}-\frac{1}{2}-\frac{1}{2} \cosh \frac{2 x}{a} ight) d x \& =4 \pi a\left[2 a \sinh \frac{x}{a}-\frac{x}{2}-\frac{a}{4} \sinh \frac{2 x}{a} ight]_0^b \& =\pi a\left(8 a \sinh \frac{b}{a}-2 b-a \sinh \frac{2 b}{a} ight) .\end{aligned}[/latex]

The area, [latex]A_2[/latex], of the two flat ends is given by

[latex]\begin{aligned}A_2 & =2 \pi a^2\left(2-\cosh \frac{b}{a} ight)^2 \& =2 \pi a^2\left(4-4 \cosh \frac{b}{a}+\cosh ^2 \frac{b}{a} ight) .\end{aligned}[/latex]

And so the total area is

[latex]\begin{aligned}A= & \pi a\left[4 a\left(e^{b / a}-e^{-b / a} ight)-2 b-\frac{a}{2}\left(e^{2 b / a}-e^{-2 b / a} ight) ight. \& \left.+8 a-4 a\left(e^{b / a}+e^{-b / a} ight)+\frac{2 a}{4}\left(e^{2 b / a}+2+e^{-2 b / a} ight) ight] \= & \pi a\left(9 a-8 a e^{-b / a}+a e^{-2 b / a}-2 b ight).\end{aligned}[/latex]

Question 3.27: A closed barrel has as its curved surface the surface obtain...

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