 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-4/11g/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 .

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 .

\int_{R} K d A = - 2 π (1 - a) .

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