Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 12D / a / i

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- Solution i

Solution

For a partition

P : 0 = x_{0} < x_{1} < \dots < x_{m} = 1,

put

M_{j} = sup_{[x_{j - 1}, x_{j}]} f, m_{j} = in f_{[x_{j - 1}, x_{j}]} f .

The upper and lower Darboux sums are

U (f, P) = \sum_{j = 1}^{m} M_{j} (x_{j} - x_{j - 1}), L (f, P) = \sum_{j = 1}^{m} m_{j} (x_{j} - x_{j - 1}) .

The upper and lower Darboux integrals are

\overline{\int_{0}^{1}} f = in f_{P} U (f, P), \underline{\int_{0}^{1}} f = sup_{P} L (f, P) .

The bounded function is Riemann integrable when these two numbers agree; their common value is its integral.

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