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This was one of the easiest problems. The volume up to point x is simply

V (x) = ∫_{t= xlow}^x P( x )dx = ∑_{i =0}ⁿ a_i/(i + 1) · (x_i+1 − x_low)ⁱ⁺¹,

and to find the value of x for which V ( x ) is some multiple of inc one can just binary search.