to i.left :fn :x :step
  output invoke :fn :x
end
to i.right :fn :x :step
  output invoke :fn :x + :step
end
to i.mid :fn :x :step
  output invoke :fn :x + :step/2
end
to i.trapezium :fn :x :step
  output ((i.left :fn :x :step) + (i.right :fn :x :step)) / 2
end
to i.simpsons :fn :x :step
  output ( (i.left :fn :x :step)
         + (i.mid :fn :x :step) * 4
         + (i.right :fn :x :step) ) / 6
end

to integrate :method :fn :steps :a :b
  localmake "step (:b - :a) / :steps
  localmake "sigma 0
  ; for [x :a :b-:step :step] [make "sigma :sigma + apply :method (list :fn :x :step)]
  repeat :steps [
    make "sigma :sigma + (invoke :method :fn :a :step)
    make "a :a + :step ]
  output :sigma * :step
end

to fn2 :x
  output 2 / (1 + 4 * :x * :x)
end
print integrate "i.left      "fn2 4 -1 2  ; 2.456897
print integrate "i.right     "fn2 4 -1 2  ; 2.245132
print integrate "i.mid       "fn2 4 -1 2  ; 2.496091
print integrate "i.trapezium "fn2 4 -1 2  ; 2.351014
print integrate "i.simpsons  "fn2 4 -1 2  ; 2.447732