; integrate : (num -> num) num num num -> num
; Numerically integrates the given function between xlo and xhi.
; Approximation accuracy is based upon threshold -- the smaller the
; threshold, the more accurate the result.
; Assumes xlo < xhi.
(define (integrate f xlo xhi threshold)
  (local [; avg2 : num num -> num
	  ; Returns the average of the two inputs.
	  (define (avg2 a b)
	    (/ (+ a b) 2))

	  ; area : num num -> num
	  ; Returns the approximate area under the function's curve
	  ; between xlo and xhi.
	  ; Assumes xlo < xhi.
	  (define (area xlo xhi)
	    ; Approximate by trapezoid method.
	    (* (avg2 (f xlo) (f xhi))
	       (- xhi xlo)))


	  ; integrate-help : num num -> num
	  ; A specialized version of integrate that uses the
	  ; original function and threshold.
	  (define (integrate-help xlo xhi)
	    (cond
	     [(< (- xhi xlo) threshold)
	      (area xlo xhi)]
	     [else
	      (local [(define xmid (/ (+ xhi xlo) 2))]
		     (+ (integrate-help xlo xmid)
			(integrate-help xmid xhi)))]))]
	 (integrate-help xlo xhi)))