Integral Series

Calculus Level 5

Let \(f(x) \) be a continuous function defined over \( [0, \infty) \) such that \( \displaystyle \int_0^{\infty} f(x)\ dx = 63 \pi \). Furthermore, for any integer \(n \geq 1\), the expression

\[ \dfrac{ \large \displaystyle \int_{n\alpha}^{(n+1)\alpha} \! f(x) \: \mathrm{d}x }{ \large \displaystyle \int_{(n-1)\alpha}^{n\alpha} \! f(x) \: \mathrm{d}x } \]

is constant given any positive real number \(\alpha\) kept fixed. If \( \displaystyle \int_0^1 f(x)\ dx = 7 \pi \), find \( \lfloor 1000 \times f(0) \rfloor \).

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