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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 26I / b / i / Solution

Put

h (x) = u_{+} (x) - u_{-} (x) .

Vertical slicing and Fubini's theorem give

A (Ω) = \int_{x_{0}}^{x_{1}} h (x) d x .

The vertical slice of

S_{L} (Ω)

is

(- h (x) /2, h (x) /2)

and has the same length

h (x)

. Therefore

A (S_{L} (Ω)) = \int_{x_{0}}^{x_{1}} h (x) d x = A (Ω) .

Solved by gpt-5.6-sol high.

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