Lec_05_integral.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Physics 300 \n",
    "## Computational Physics I (Fall 2018)\n",
    "## BPB-248, Mon/Wed 02:30 - 03:45 pm \n",
    "\n",
    "|Instructor| Prof. Qiang Zhu|\n",
    "|--|-------------------------------|\n",
    "|Email | qiang.zhu@unlv.edu|\n",
    "|Website|http://www.physics.unlv.edu/~qzhu/|\n",
    "|Office| BPB 232|\n",
    "|Office hours | Mon/Wed 03:45 - 05:00 pm |\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5 Integrals\n",
    "In the following lecture, we will start to learn about the ideas of computational physics and \n",
    "apply the python programing to solve the real problems.\n",
    "\n",
    "One of the most basic but also most important applications of computers\n",
    "in physics is the evaluation of integrals and derivatives.\n",
    "\n",
    "As we have learned in the calculus class, we can derive the analytical integrals for a lot of functions \n",
    "such as linear, quadratic, ploynomial, triangle functions. But most of functions cannot. \n",
    "__They can, however, almost always be done on a computer.__\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's start with a simple example, try to do integration on x^2 over [0,3]\n",
    "%matplotlib inline  \n",
    "\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "x = np.linspace(0,3,100)\n",
    "y = x*x\n",
    "plt.plot(x,y)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('$x^2$')\n",
    "plt.xlim([0,3])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Mathematically, the integral means the area under the curve\n",
    "x = np.linspace(0,3,100)\n",
    "y = x*x\n",
    "y0 = np.zeros(100)\n",
    "plt.plot(x,y)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('$x^2$')\n",
    "plt.xlim([0,3])\n",
    "plt.fill_between(x, y0, y, where=y>=y0, facecolor='green')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mathematically, the integral means the area under the curve\n",
    "\n",
    "Suppose we have a function $f(x)$ and we want to calculate its integral with\n",
    "respect to $x$ from $x$ = a to $x$ = b, which we denote I(a, b):\n",
    "\n",
    "$$ I(a,b) = \\int_a^b f(x)dx $$\n",
    "\n",
    "This is equivalent to calculating the area under the curve of f(x) from a to b.\n",
    "There is no known way to calculate such an area exactly in all cases on a computer.\n",
    "\n",
    "- how to estimate the area?\n",
    "- how to make it as accurate as possible?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.1 Rectangle rules\n",
    "\n",
    "The most straightforward way is to:\n",
    "- divide the area up into rectangular slices, \n",
    "- calculate the area of each one\n",
    "- add them up. \n",
    "\n",
    "This, however, is a pretty poor approximation. The area\n",
    "under the rectangles is not very close to the area under the curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def Rectangle(start, end, parts, plot=1):\n",
    "    \"\"\"\n",
    "    Rectrangle sum rule\n",
    "    \n",
    "    \"\"\"\n",
    "    #define the function\n",
    "    f = lambda x: x * x  \n",
    "    \n",
    "    #define the X,Y points\n",
    "    deltax = (end - start) / parts\n",
    "    resultsx = np.linspace(start, end, parts)\n",
    "    resultsy = f(resultsx)\n",
    "    \n",
    "    # To calculate the area\n",
    "    area = np.empty([parts], float)\n",
    "    for i in range(parts):\n",
    "            area[i] = resultsy[i] * deltax\n",
    "    \n",
    "\n",
    "    # By default, we also output the plot.\n",
    "    if plot==1:\n",
    "        x = np.linspace(start,end,100)\n",
    "        y = x*x        \n",
    "        plt.plot(x, y, 'r')\n",
    "        #plt.xlim([start,end])\n",
    "        plt.bar(resultsx+deltax/2, resultsy, deltax, edgecolor ='black')\n",
    "        #plt.bar(resultsx, resultsy, deltax, edgecolor ='black')\n",
    "        plt.show()\n",
    "        print(\"The Sum of the area is: \", sum(area))\n",
    "    \n",
    "    return sum(area)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6 27 3\n"
     ]
    }
   ],
   "source": [
    "f1 = lambda x, y: x * y  \n",
    "f2 = lambda x: x * x * x   \n",
    "f3 = lambda y: y-1  \n",
    "\n",
    "print(f1(2,3), f2(3), f3(4))\n",
    "np.linspace(start, end, parts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.,  0.],\n",
       "       [ 0., -1.,  0.],\n",
       "       [ 0.,  0.,  1.]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.empty([3,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Sum of the area is:  9.045454545454543\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9.045454545454543"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rectangle(0,3,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Sum of the area is:  9.236842105263156\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9.236842105263156"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rectangle(0,3,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.2 The Trapezoidal rule\n",
    "\n",
    "The rectangle approximation is easy to calculate. This, however, is a pretty poor approximation. The area\n",
    "under the rectangles is not very close to the area under the curve.\n",
    "\n",
    "A better approach is to use the trapezoids rather than rectangles.\n",
    "The area under the trapezoids is a considerably better approximation to the\n",
    "area under the curve, and this approach, though simple, often gives perfectly\n",
    "adequate results.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def Trapezoid(start, end, parts, plot=1):\n",
    "    \"\"\"\n",
    "    Trapezoid sum rule\n",
    "    \n",
    "    \"\"\"\n",
    "    #define the function\n",
    "    f = lambda x: x * x  \n",
    "    \n",
    "    #define the X, Y points \n",
    "    deltax = (end - start) / parts\n",
    "    resultsx = np.linspace(start, end, parts+1)\n",
    "    resultsy = f(resultsx)\n",
    "\n",
    "    # To calculate the area\n",
    "    area = np.empty([parts], float)\n",
    "    for i in range(parts):\n",
    "            area[i] = (resultsy[i]+resultsy[i+1]) * deltax/2\n",
    "\n",
    "    # By default, we also output the plot.\n",
    "    if plot==1:\n",
    "        x = np.linspace(start,end,100)\n",
    "        y = f(x)        \n",
    "        plt.plot(x, y, 'r')\n",
    "        plt.xlim([start,end])\n",
    "        \n",
    "        y2 = np.array([0,0])\n",
    "        for i in range(parts):\n",
    "            x0 = resultsx[i:i+2]\n",
    "            y1 = resultsy[i:i+2]\n",
    "            plt.fill_between(x0, y1, y2, where=y1>=y2, facecolor='blue')\n",
    "            linex, liney = [resultsx[i+1], resultsx[i+1]], [0, resultsy[i+1]]\n",
    "            plt.plot(linex, liney, color='black', linewidth=2.0)        \n",
    "        \n",
    "        plt.show()\n",
    "        print(\"The Sum of the area is: \", sum(area))\n",
    "    \n",
    "    return sum(area)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Sum of the area is:  9.18\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9.18"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Trapezoid(0,3,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Sum of the area is:  9.045\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9.045"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Trapezoid(0,3,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.3 The Simpson’s rule\n",
    "\n",
    "The trapezoidal rule is simple, taking only a few lines of code as we have seen. \n",
    "And it is often perfectly adequate for calculations where no great accuracy is required. \n",
    "It happens frequently in physics calculations that we don’t need an answer accurate to many significant\n",
    "figures and in such cases the ease and simplicity of the trapezoidal rule can\n",
    "make it the method of choice. \n",
    "\n",
    "However, there are also cases where greater accuracy is required. \n",
    "\n",
    "As we have seen we can increase the accuracy of the trapezoidal rule by _increasing the number N of steps used in the calculation_. \n",
    "\n",
    "But in some cases, _the calculation can become slow_.\n",
    "\n",
    "There are other, more advanced schemes for calculating integrals\n",
    "that can achieve high accuracy while still arriving at an answer quickly. \n",
    "\n",
    "We can often get a better result if we approximate the function instead with curves of some kind.\n",
    "\n",
    "Simpson’s rule does exactly this, using quadratic curves.\n",
    "\n",
    "In order to specify a quadratic completely \n",
    "\n",
    "__one needs three points, not just two as with a straight line.__\n",
    "\n",
    "Suppose, as before, that our integrand is denoted f(x) and the spacing of\n",
    "adjacent points is $h$. And suppose for the purposes of argument that we have\n",
    "three points at $x$ = −h, 0, and +h. If we fit a quadratic $Ax^2 + Bx + C$ through\n",
    "these points, then by definition we will have:\n",
    "\n",
    "$$f(-h) = Ah^2 - Bh + C$$\n",
    "$$f(0) = C$$ \n",
    "$$f(h) = Ah^2 + Bh + C$$\n",
    "\n",
    "Solving these equations gives\n",
    "\n",
    "$$ A = \\frac{1}{h^2}[\\frac{f(-h)}{2} - f(0) + \\frac{f(h)}{2}]$$\n",
    "$$ B=\\frac{1}{2h}[f(h)-f(-h)] $$ \n",
    "$$ C = f(0)$$\n",
    "\n",
    "and the area under the curve of f(x) from −h to +h is given approximately by\n",
    "the area under the quadratic:\n",
    "\n",
    "$$ \\int^h_{-h} (Ax^2 + Bx + C) dx = \\frac{2}{3}Ah^3 + 2Ch = \\frac{1}{3}h[f(-h)+4f(0)+f(h)]$$\n",
    "\n",
    "This is _Simpson’s_ rule. It gives us an approximation to the area under two adjacent\n",
    "slices of our function. Note that the final formula for the area involves\n",
    "only h and the value of the function at evenly spaced points, just as with the\n",
    "trapezoidal rule. So to use Simpson’s rule we don’t actually have to worry\n",
    "about the details of fitting a quadratic—we just plug numbers into this formula\n",
    "and it gives us an answer.\n",
    "\n",
    "\n",
    "Applying Simpson’s rule involves dividing\n",
    "the domain of integration into many slices and using the rule to separately\n",
    "estimate the area under successive pairs of slices, then adding the estimates\n",
    "for all pairs to get the final answer. If, as before, we are integrating from $x$ = a\n",
    "to $x$ = b in slices of width $h$ then the three points bounding the first pair of\n",
    "slices fall at $x = a, a + h, a + 2h$, those bounding the second pair at $a + 2h,\n",
    "a + 3h, a + 4h$, and so forth. Then the approximate value of the entire integral\n",
    "is given by\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "I(a,b) \\approx & \\frac{1}{3}h[f(a)+4f(a+h)+f(a+2h)] \\\\\n",
    "               & \\frac{1}{3}h[f(a+2h)+4f(a+3h)+f(a+4h)] \\\\\n",
    "               &  ...\\\\\n",
    "               &  ...\\\\\n",
    "               & \\frac{1}{3}h[f(a+(N-2)h)+4f(a+(N-1)h)+f(b)] \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "see more details in: https://en.wikipedia.org/wiki/Simpson%27s_rule"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def Simpson(start, end, parts, plot=1):\n",
    "    \"\"\"\n",
    "    Simpson sum rule\n",
    "    \n",
    "    \"\"\"\n",
    "    #define the function\n",
    "    f = lambda x: x**4 - 2*x + 1  \n",
    "    \n",
    "    #define the X, Y points \n",
    "    deltax = (end - start) / parts\n",
    "    resultsx = np.linspace(start, end, parts+1)\n",
    "    resultsy = f(resultsx)\n",
    "\n",
    "    #-----------------------------------------------------\n",
    "    #define the Simpson points and calculate the area here\n",
    "    #-----------------------------------------------------\n",
    "    area = []\n",
    "\n",
    "    #By default, we also output the plot.\n",
    "    if plot==1:\n",
    "        x = np.linspace(start,end,100)\n",
    "        y = f(x)        \n",
    "        plt.plot(x, y, 'r')\n",
    "        \n",
    "        #-----------------------------------------------------\n",
    "        #draw the curves based on Simpson points here\n",
    "        #-----------------------------------------------------             \n",
    "    \n",
    "        plt.xlim([start,end])\n",
    "        plt.ylim([min(y),max(y)])\n",
    "        plt.show()\n",
    "    \n",
    "    return sum(area)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Simpson(0, 2, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.4 Futher expansion to higher order: \n",
    "As we have seen, the trapezoidal rule is based on approximating an integrand\n",
    "$f(x)$ with straight-line segments, while Simpson’s rule uses quadratics.\n",
    "We can create higher-order (and hence potentially more accurate) rules by using\n",
    "higher-order polynomials, fitting $f(x)$ with cubics, quartics, and so forth.\n",
    "The general form of the trapezoidal and Simpson rules is\n",
    "\n",
    "$$\\int_a^b f(x)dx \\approx \\sum_{k=1}^{N}w_kf(k)$$\n",
    "\n",
    "where the $x_k$ are the positions of the sample points at which we calculate the\n",
    "integrand and the $w_k$ are some set of weights. \n",
    "\n",
    "- In the trapezoidal rule, the first and last weights are 1/2 and the others are all 1,\n",
    "- In Simpson’s rule the weights are 1/3 for the first and last slices and alternate between 4/3 and 2/3 for the other slices\n",
    "- What will happen on cubic?\n",
    "\n",
    "|Order| Apporximation | Coefficients   |\n",
    "|-----|:-------------:|--------------------------------------------------:|\n",
    "|0    | rectangle     |   1,1,........1,1|\n",
    "|1    | straight line |1/2,1,1,.....1,1/2|\n",
    "|2    | quadratic     |1/3, 4/3, 2/3, ..,4/3,1/3|\n",
    "|3    | cubic         |...................|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 5.5 Homework: \n",
    "Calculate the following integral based on three different rules\n",
    "\n",
    "$$\\int_0^3 (x^2-x+1 dx)$$\n",
    "\n",
    "$$\\int_0^3 (x^4-x+1 dx)$$\n",
    "\n",
    "$$\\int_0^3 \\text{exp}(-x^2) dx$$\n",
    "\n",
    "For the 1st one, how many sampling points do you need for Simpson rule?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.6 Further reading on uneven sampling and Gaussian Quardrature\n",
    "\n",
    "We have been assuming here that the sample points are evenly spaced. \n",
    "This is easy to program. However, it is also possible to derive integration methods with unevenly spaced points.\n",
    "Sometimes, they can give very accurate answers with only small numbers of points.\n",
    "One of the well known methods called __Gaussian Quardrature__.\n",
    "\n",
    "### Project 2: \n",
    "Write a wiki page to introduce Gaussian Quardrature from the foundamental of mathematics, and give some applications"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8.666666666666684\n"
     ]
    }
   ],
   "source": [
    "from numpy import ones,copy,cos,tan,pi,linspace\n",
    "\n",
    "def gaussxw(N):\n",
    "\n",
    "    # Initial approximation to roots of the Legendre polynomial\n",
    "    a = linspace(3,4*N-1,N)/(4*N+2)\n",
    "    x = cos(pi*a+1/(8*N*N*tan(a)))\n",
    "\n",
    "    # Find roots using Newton's method\n",
    "    epsilon = 1e-15\n",
    "    delta = 1.0\n",
    "    while delta>epsilon:\n",
    "        p0 = ones(N,float)\n",
    "        p1 = copy(x)\n",
    "        for k in range(1,N):\n",
    "            p0,p1 = p1,((2*k+1)*x*p1-k*p0)/(k+1)\n",
    "        dp = (N+1)*(p0-x*p1)/(1-x*x)\n",
    "        dx = p1/dp\n",
    "        x -= dx\n",
    "        delta = max(abs(dx))\n",
    "\n",
    "    # Calculate the weights\n",
    "    w = 2*(N+1)*(N+1)/(N*N*(1-x*x)*dp*dp)\n",
    "\n",
    "    return x,w\n",
    "\n",
    "\n",
    "def f(x):\n",
    "    return x**5 - 2*x + 1\n",
    "\n",
    "N = 3\n",
    "a,b = 0.0, 2.0\n",
    "\n",
    "# Calculate the sample points and weights, then map them\n",
    "# to the required integration domain\n",
    "x,w = gaussxw(N)\n",
    "xp = 0.5*(b-a)*x + 0.5*(b+a)\n",
    "wp = 0.5*(b-a)*w\n",
    "\n",
    "# Perform the integration\n",
    "s = 0.0\n",
    "for k in range(N):\n",
    "    s += wp[k]*f(xp[k])\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Appendix \n",
    "## Code for Simpson Rule\n",
    "\n",
    "```python\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def Simpson1(start, end, parts):\n",
    "    \"\"\"\n",
    "    Rectrangle sum rule\n",
    "    \n",
    "    \"\"\"\n",
    "    #define the function\n",
    "    f = lambda x: x * x  \n",
    "    \n",
    "    #define the X,Y points\n",
    "    deltax = (end - start) / parts\n",
    "    resultsx = np.linspace(start, end, parts+1)\n",
    "    #resultsx = np.append(resultsx, end+deltax)\n",
    "    resultsy = f(resultsx)\n",
    "    \n",
    "    # To calculate the area\n",
    "    area = np.empty([parts+1], float)\n",
    "    for i in range(parts+1):\n",
    "        if i == 0 or i == parts:\n",
    "            coef = 1/3\n",
    "        elif i%2 == 1:\n",
    "            coef = 4/3\n",
    "        else: \n",
    "            coef = 2/3\n",
    "        area[i] = coef*resultsy[i]*deltax\n",
    "        print(resultsx[i], resultsy[i], coef)\n",
    "\n",
    "    \n",
    "    return sum(area)\n",
    "```\n",
    "\n",
    "## Built-in function in scipy\n",
    "\n",
    "https://docs.scipy.org/doc/scipy/reference/integrate.html#module-scipy.integrate\n",
    "\n",
    "```python\n",
    "from scipy import integrate\n",
    "f = lambda x: x * x  \n",
    "integrate.quad(f, 1, 3)    \n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def Simpson1(start, end, parts):\n",
    "    \"\"\"\n",
    "    Simpson sum rule\n",
    "\n",
    "    \"\"\"\n",
    "    #define the function\n",
    "    f = lambda x: x * x * x\n",
    "\n",
    "    #define the X,Y points\n",
    "    deltax = (end - start) / parts\n",
    "    resultsx = np.linspace(start, end, parts+1)\n",
    "    #resultsx = np.append(resultsx, end+deltax)\n",
    "    resultsy = f(resultsx)\n",
    "\n",
    "    # To calculate the area\n",
    "    area = np.empty([parts+1], float)\n",
    "    for i in range(parts+1):\n",
    "        if i == 0 or i == parts:\n",
    "            coef = 1/3\n",
    "        elif i%2 == 1:\n",
    "            coef = 4/3\n",
    "        else: \n",
    "            coef = 2/3\n",
    "        area[i] = coef*resultsy[i]*deltax\n",
    "        #print(resultsx[i], resultsy[i], coef)\n",
    "\n",
    "\n",
    "    return sum(area)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2500.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Simpson1(0, 10, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2500.0000000000005"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Simpson1(0, 10, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2500.0000000000014"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Simpson1(0, 10, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(0,10, 100)\n",
    "plt.plot(x, np.sin(x)*np.exp(x) + 11)\n",
    "plt.ylim([0, 4000])\n",
    "plt.xlim([0, 8])\n",
    "\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}