Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 11G / b

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- Solution b

Solution

The area element is

d A = f 1 + f^{'2} d x d θ .

Consequently

K d A = - \frac{f ^{''}}{( 1 + f ^{'2} ) ^{3/2}} d x d θ = - d (\frac{f ^{'}}{1 + f ^{'2}}) d θ .

At

x = 2 - a

, smooth matching to the cylinder gives

f^{'} = 0

. At

x = 2 + a

, the spherical formula gives

f^{'} = \frac{1 - a}{2 a - a ^{2}}, \frac{f ^{'}}{1 + f ^{'2}} = 1 - a .

The total Gaussian curvature of a surface-of-revolution strip is therefore

\int_{R} K d A = - 2 π (1 - a) .

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