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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.

Advent21
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  • Why would you want to do Monte Carlo calculations without a computer? That seems very strange. – Arthur Apr 27 '20 at 07:43
  • @Arthur When we have tests, our professor does not allow a computer. It's all pencil and paper. – Advent21 Apr 27 '20 at 07:45
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    In those circumstances, your professor can't really test your ability to do Monte Carlo calculations, because the ability to make thousands of random numbers is key, and you can't do that with pen and paper on a test. So it would be counterproductive of them to ask you to do it. Maybe they want you to write down an algorithm, not actually do calculations? – Arthur Apr 27 '20 at 07:47
  • @Arthur you may be correct. I guess I'll see when test day comes. If that's the case, I will feel a lot better – Advent21 Apr 27 '20 at 07:50

1 Answers1

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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.

uniquesolution
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