type Partition =
  { points : double []
  ; deltaX : double
  }



let parition lo hi n =
  let dx = (hi - lo) / double n
  { points = [|for i = 0 to n do
                 yield lo + dx * double i
             |]
  ; deltaX = dx
  }



let lowerSum f p =
  Array.pairwise p.points
  |> Array.sumBy (fun (l, r) -> min (f l) (f r) * p.deltaX)



let upperSum f p =
  Array.pairwise p.points
  |> Array.sumBy (fun (l, r) -> max (f l) (f r) * p.deltaX)



let main =
  // f(x) = x^3 - 3x^2 - x + 3
  let f x = (x ** 3.0) - 3.0 * (x ** 2.0) - x + 3.0
  // [-2, 3]
  let (lo, hi) = -2.0, 3.0

  let n = 10

  let p = parition lo hi n

  printfn "P: %A" p
  printfn "L(f,P) = %f" (lowerSum f p)
  printfn "U(f,P) = %f" (upperSum f p)