Methods | Part IB, 2001

A curve y(x)y(x) in the xyx y-plane connects the points (±a,0)(\pm a, 0) and has a fixed length l,2a<l<πal, 2 a<l<\pi a. Find an expression for the area AA of the surface of the revolution obtained by rotating y(x)y(x) about the xx-axis.

Show that the area AA has a stationary value for

y=1k(coshkxcoshka),y=\frac{1}{k}(\cosh k x-\cosh k a),

where kk is a constant such that

lk=2sinhka.l k=2 \sinh k a .

Show that the latter equation admits a unique positive solution for kk.

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