Monte-Carlo-Integrals.ipynb

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   "source": [
    "# Monte-Carlo integral Approximation\n",
    "import numpy as np"
   ]
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   "source": [
    "##### We want to approzimate the following integral with Monte Carlo Simulations\n",
    "$$\\int_{3}^{5} e^{x^2} dx = $$\n",
    "##### Using the transformation\n",
    "$$ y = \\frac{x-a}{b-a}$$"
   ]
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     "text": [
      "7326219978.280555\n"
     ]
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    "# Integration limits\n",
    "a = 3\n",
    "b = 5\n",
    "\n",
    "# Number of simulations\n",
    "N = 1000000\n",
    "\n",
    "# Algorithm\n",
    "suma = 0\n",
    "for i in range(N):\n",
    "    x = np.random.uniform()*(b-a)+a\n",
    "    suma = suma + np.exp(x**2)\n",
    "E = suma/N\n",
    "R = E*(b-a)\n",
    "print(R)"
   ]
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   "source": [
    "##### We see that as the number of sumulations N goes up, we approximate the real value of the integral\n",
    "$$\\int_{3}^{5} e^{x^2} dx = 7354152303 $$"
   ]
  },
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   "source": [
    "##### We will now approximate this integral using the same transformation\n",
    "$$\\int_{0}^{1} \\frac{1}{\\sqrt{1+sin(x)^2}} dx =$$\n",
    "$$ y = \\frac{x-a}{b-a}$$"
   ]
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     "text": [
      "0.8964447155243265\n"
     ]
    }
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   "source": [
    "# Integration limits\n",
    "a = 0\n",
    "b = 1\n",
    "\n",
    "# Number of simulations\n",
    "N = 1000000\n",
    "\n",
    "# Algorithm\n",
    "suma = 0\n",
    "for i in range(N):\n",
    "    x = np.random.uniform()*(b-a)+a\n",
    "    suma = suma + 1/(np.sqrt(1+(np.sin(x)**2)))\n",
    "E = suma/N\n",
    "R = E*(b-a)\n",
    "print(R)"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "##### We see that it also approximates its real value\n",
    "$$\\int_{0}^{1} \\frac{1}{\\sqrt{1+sin(x)^2}} dx = 0.896393$$"
   ]
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