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Given:

(define (monte-carlo trials experiment) (define (iter trials-remaining trials-passed) (cond ((= trials-remaining 0) (/ trials-passed trials)) ((experiment) (iter (- trials-remaining 1) (+ trials-passed 1))) (else (iter (- trials-remaining 1) trials-passed)))) (iter trials 0)) (define (random-in-range low high) (let ((range (- high low))) (+ low (random range))))

The solution is:

(define (P x y) (< (+ (expt (- x 5) 2) (expt (- y 7) 2)) (expt 3 2))) (define (estimate-integral P x1 x2 y1 y2 trials) (define (experiment) (P (random-in-range x1 x2) (random-in-range y1 y2))) (monte-carlo trials experiment))

Test:

```
(estimate-integral P 2.0 8.0 4.0 10.0 100)
```

Then we can estimate pi with the fact that a circle area is (pi * r²).

Hence pi ≅ (Monte Carlo results * rectangle area) / r²

```
(define pi-approx
(/ (* (estimate-integral P 2.0 8.0 4.0 10.0 10000) 36)
9.0))
pi-approx
```

Which gave 3.1336 during my test.

This function has to be tested under MIT Scheme, neither gambit-scheme or SISC implements (random) - actually (random) is not part of R5RS nor SRFI.

NB: using 2.0 instead of 2 in estimate-integral is primordial. If you pass two integers to (random-in-range low high), it will return another integer strictly inferior to your 'high' value — and this completely screws the Monte-Carlo method (it then estimates pi to ~3.00).

Using Racket's built-in (random), which returns a float between 0 and 1, the following set of functions can be used to find the numerical integral of a function with a predicate, using the provided (monte-carlo) function:

(define (random-in-range low high) (let ([range (- high low)]) (+ low (* (random) range)))) (define (get-area lower-x lower-y upper-x upper-y) (* (- upper-x lower-x) (- upper-y lower-y))) (define (estimate-integral pred lower-x lower-y upper-x upper-y trials) (let ([area (get-area lower-x lower-y upper-x upper-y)] [experiment (lambda () (pred (random-in-range lower-x upper-x) (random-in-range lower-y upper-y)))]) (* area (monte-carlo trials experiment))))

Testing with a large number of trials:

```
(estimate-integral (lambda (x y) (<= (+ (* x x) (* y y)) 1.0)) -1.0 -1.0 1.0 1.0 100000000)
3.14130636
```

If you're using Racket the random function doesn't work as described in the text. You might want to use this instead:

```
(define (random-in-range low high)
(let ((range (- high low)))
(+ low (* (random) range))))
```

Gambit has (random-real) for a random inexact real between 0 and 1 and (random-integer n) for a random integer between 0 and n.

I think the procedure estimate-integral is not very nicely formulated in the text above, in my opinion, a better solution for this exercise would be:

We should leave the job of generating random numbers to the predicate, instead of intertwining it with estimate-integral. If we do it like this, we cannot adapt estimate-integral to other predicates.