How do you do Monte Carlo without computer programs?

Question

I am trying to approximate the area under the curve of $f(x) = \sqrt{x}$ over the integral $1/2 ≤ x ≤ 3/2$. I am so confused on how to use Monte Carlo without the use of a computer program to estimate the area. I integrated $f(x)$ but that gives me the actual area. I tried making a rectangle around the curve, but how would you find the y values without using a calculator/computer? I've looked at several videos and other forums/questions. At this point, I am so confused. I need guidance, please.

Answer 1

One way to do it is as follows.

1. Get a large sand-box.
2. Inside the sand-box, draw a sufficiently large enough square whose (scaled) dimensions are $[0,\frac{3}{2}]\times [0,\frac{3}{2}]$
3. Carefully draw the graph of $\sqrt{x}$ within the square.
4. Get a lot of marbles.
5. Step back, and start throwing marbles into the square. Get some help, if necessary.
6. After you've thrown many marbles, count how many fell inside the area you are interested in. The proportion to the total number of marbles thrown is a Monte-Carlo estimate of the ratio between the desired integral to the area of the square.

Obviously, it is much easier to do Monte-Carlo with a computer.