right_rect(e, x, a, b, n) := block([h: (b - a) / n, s: 0],
   for i from 1 thru n do s: s + subst(x = a + i * h, e),
   s * h)$

left_rect(e, x, a, b, n) := block([h: (b - a) / n, s: 0],
   for i from 1 thru n do s: s + subst(x = a + (i - 1) * h, e),
   s * h)$

mid_rect(e, x, a, b, n) := block([h: (b - a) / n, s: 0],
   for i from 1 thru n do s: s + subst(x = a + (i - 1/2) * h, e),
   s * h)$

trapezium(e, x, a, b, n) := block([h: (b - a) / n, s: 0],
   for i from 1 thru n - 1 do s: s + subst(x = a + i * h, e),
   ((subst(x = a, e) + subst(x = b, e)) / 2 + s) * h)$

simpson(e, x, a, b, n) := block([h: (b - a) / n, s: 0],
   for i from 1 thru n do
      s: s + subst(x = a + i * h, e) + 2 * subst(x = a + (i - 1/2) * h, e),
   (subst(x = a, e) - subst(x = b, e) + 2 * s) * h / 6)$

/* some tests */

simpson(log(x), x, 1, 2, 20), bfloat;
2 * log(2) - 1 - %, bfloat;

trapezium(1/x, x, 1, 100, 10000) - log(100), bfloat;