Minimizing Resultant Volume

Calculus Level pending

Let \(a\), \(b\), and \(c\) be real constants. Minimize the volume of the region bounded between \(y = x^3 + ax^2 +bx + c\), \(x= 0\) and \(x=1\), when it is revolved about the \(x\)-axis.

If this volume can be expressed as \(\dfrac mn \pi\), where \(m\) and \(n\) are coprime positive integers, submit your answer as \(m+n\).

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