# 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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