{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Definite Integrals, Part 1: The Building Blocks\n",
    "\n",
    "**Expanded on Monday, March 8,** adding an Appendix on the [Left-hand Endpoint Rule](#Left-hand-Rule)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Sections 5.2.1, 5.2.3 and 5.2.4 of [Sauer](../references.html#Sauer)\n",
    "- Sections 4.3 and 4.4 of [Burden&Faires](../references.html#Burden-Faires)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "The objective of this and several subsequent sections is to develop methods for approxmating a definite integral\n",
    "\n",
    "$$ I = \\int_a^b f(x) \\; dx $$\n",
    "\n",
    "This is arguably even more important than approximating derivatives, for several reasons;\n",
    "in particualr, because there are many functions for which antiderivative formulas cannot be found,\n",
    "so that the result of the Fundamental Theorem of Calculus, that\n",
    "\n",
    "$$ \\int_a^b f(x) \\; dx = F(b) - F(a), \\text{ for } F \\text{ any antiderivative of } f$$\n",
    "\n",
    "does not help us.\n",
    "\n",
    "One core idea is to approximate the function $f$ by a polynomial (or several), and use its integral as an approximation.\n",
    "The two simplest possibilities here are approximating by a constant and by a straight line;\n",
    "here we explore the latter; th former wil be visited soon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import array, linspace, exp\n",
    "from matplotlib.pyplot import figure, plot, title, grid, legend"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Approximating with a single linear function: the Trapezoid Rule\n",
    "\n",
    "The idea is to approximate $f:[a, b] \\to \\Bbb{R}$ by collocation at the end points of this interval:\n",
    "\n",
    "$$ f(x) \\approx L(x) := \\frac{f(a)(b-x) + f(b)(x-a)}{b-a}, = f_{ave} (b-a) $$\n",
    "\n",
    "Then the approximation — which will be called $T_1$, for reasons that will becom clear soon — is\n",
    "\n",
    "$$I \\approx T_1 = \\int_a^b L(x) dx = \\frac{f(a) + f(b)}{2} (b-a) $$\n",
    "\n",
    "This can be interpreted as replacing $f(x)$ by $f_{ave}$ the average of the value at the end points,\n",
    "and inegrting that simple function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the example $f(x) = e^x$ on $[-1, 3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 1\n",
    "b = 3\n",
    "def f(x): return exp(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_ave = (f(a) + f(b))/2\n",
    "x = linspace(a, b)\n",
    "figure(figsize=[12, 8])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f(a), f(b), 0, 0], label=\"Trapezoid Rule\")\n",
    "plot([a, a, b, b, a], [0, f_ave, f_ave, 0, 0], '-.', label=\"Trapezoid Rule area\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximation $T_1$ is the area of the orange trapezoid (hence the name!) which is also the area of the green rectangle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Approximating with a constant: the Midpoint Rule\n",
    "\n",
    "The idea here is to approximate $f:[a, b] \\to \\Bbb{R}$ by its value at the midpoint of the interval,\n",
    "like the building blocks in a Riemann sum with the middel being the intuitive best choice of where to put the rectangle.\n",
    "\n",
    "$$ f(x) \\approx f_{mid} := f \\left(\\frac{a+b}{2}\\right) $$\n",
    "\n",
    "Then the approximation — which will be called $M_1$ — is\n",
    "\n",
    "$$ I \\approx M_1 = \\int_a^b f_{mid} \\, dx =  f \\left(\\frac{a+b}{2}\\right)(b-a) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the same example $f(x) = e^x$ on $[-1, 3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_mid = f((a+b)/2)\n",
    "figure(figsize=[12, 8])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f_mid, f_mid, 0, 0], 'r', label=\"Midpoint Rule\")\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximation $M_1$ is the area of the red rectangle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two methods can be compared my combining these graphs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_mid = f((a+b)/2)\n",
    "figure(figsize=[12, 8])\n",
    "plot(x, f(x))\n",
    "plot([a, a, b, b, a], [0, f(a), f(b), 0, 0], label=\"Trapezoid Rule\")\n",
    "plot([a, a, b, b, a], [0, f_ave, f_ave, 0, 0], '-.', label=\"Trapezoid Rule area\")\n",
    "plot([a, a, b, b, a], [0, f_mid, f_mid, 0, 0], 'r', label=\"Midpoint Rule\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error Formulas\n",
    "\n",
    "These graphs indicate that the trapezoid rule will over-estimate the error for this and any function that is convex up on the interval $[a, b]$.\n",
    "With closer examination it can perhaps be seen that the Midpoint Rule will instead underestimate in this situation, because\n",
    "its \"overshoot\" at left is less than its \"undershoot\" at right.\n",
    "\n",
    "We can derive error formulas that confirm this, and which are the basis for both practical error estimates and for deriving more accurate approximation methods.\n",
    "\n",
    "The first such method will be to use multiple small intervals instead of a single bigger one (using piecewise polynomial approximation) and for that, it is convenient to define $h = b-a$ which will become the parameter that we reduce in order to improve accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 1.  Error in the Trapezoid Rule, $T_1$**\n",
    "\n",
    "For a function $f$ that is twice differentiable on interval $[a, b]$, the error in the Trapezoid Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - T_1 = -\\frac{(b-a)^3}{12}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$\n",
    "\n",
    "It will be convenient to define $h := b-a$ so that this becomes\n",
    "\n",
    "$$ \\int_a^b f(x) dx - T_1 = -\\frac{h^3}{12}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 2.  Error in the Midpoint Rule, $M_1$**\n",
    "\n",
    "For a function $f$ that is twice differentiable on interval $[a, b]$ and again with $h=b-a$,\n",
    "the error in the Midpoint Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - M_1 = \\frac{h^3}{24}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These will be verified below, using the error formulas for Taylor polynomials and collocation polynomials.\n",
    "\n",
    "For now, note that:\n",
    "- The results confirm that for a function that is convex up, the Trapezoid Rule overestimates and the Midpoint Rule underestimates.\n",
    "- The ratio of the errors is approximately $-2$. This will be used to get a better result by using a weighted average: *Simpson's Rule.*\n",
    "- The errors are $O(h^3)$. This opens the door to Richardson Extrapolation, as will be seen soon in the method of *Romberg Integration.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proofs of these error results\n",
    "\n",
    "One side benefit of the following verifications is that they also offer illustrations of how the two fundamental error formulas help us: Taylor's Formula and its cousin the error formula for polynomial collocation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Proof of Theorem 1: the error in the Trapezoid Rule\n",
    "\n",
    "*The function integrated to get the Trapezoid Rule is the linear collocating polynomial $L(x)$,\n",
    "and from the section\n",
    "[Error Formulas for Polynomial Collocation](polynomial-collocation-error-formulas-python.ipynb)\n",
    "we have*\n",
    "\n",
    "$$ f(x) - L(x) = \\frac{f''(\\xi_x)}{2}(x-a)(x-b) $$\n",
    "\n",
    "*Integrating each side gives*\n",
    "\n",
    "$$\n",
    "\\int_a^b (f(x) - L(x))\n",
    "= I - T_1\n",
    "= \\int_a^b \\frac{f''(\\xi_x)}{2}(x-a)(x-b) \\, dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Remember that $\\xi_x$ depends on $x$ in an unknown way; to get around that complication,\n",
    "we introduce a result that also helps in various places later:*\n",
    "\n",
    "**Theorem 3. The Integral Mean Value Theorem**\n",
    "\n",
    "In an integral\n",
    "\n",
    "$$ \\int_a^b f(x) w(x) \\, dx  $$\n",
    "\n",
    "with $f$ continuous and the \"weight function\" $w(x)$ positive valued\n",
    "(actually, it is enough that $w(x) \\geq 0$ and it is not zero everyhere),\n",
    "there is a point $\\xi \\in [a,b]$ that gives a \"weighted average value\" for $f(x)$ in the sense that\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) w(x) \\, dx = \\int_a^b f(\\xi) w(x) \\, dx, = f(\\xi) \\int_a^b w(x) \\, dx\n",
    "$$\n",
    "\n",
    "**Proof:**\n",
    "\n",
    "*As $f$ is continuous on the closed, bounded interval $[a, b]$, the **Extreme Value Theorem** from calculus says that $f$ has a minimum $L$ and a maximum $H$ on this interval: $L \\leq f(x) \\leq H$.\n",
    "Since $w(x) \\geq 0$, this gives*\n",
    "\n",
    "$$\n",
    "L w(x) \\leq f(x) w(x) \\leq H w(x)\n",
    "$$\n",
    "\n",
    "*and by integrating,*\n",
    "\n",
    "$$\n",
    "L \\int_a^b w(x) \\,dx \\leq \\int_a^b f(x) w(x) \\,dx \\leq H \\int_a^b w(x) \\,dx\n",
    "$$\n",
    "\n",
    "*Dividing by $\\int_a^b w(x) \\,dx$ (which is positive),*\n",
    "\n",
    "$$\n",
    "L \\leq \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} \\leq H\n",
    "$$\n",
    "\n",
    "*and the **Mean Value Theorem** says that $f$ attains this value for some $\\xi \\in [L, H]$:*\n",
    "\n",
    "$$ f(\\xi) = \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} $$\n",
    "\n",
    "*Clearing the denominator gives the claimed result.*\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Returning to Theorem 1, we use this with weight function $w(x) = (x-a)(b-x), \\geq 0$ for $a \\leq x \\leq b$.\n",
    "Then with $-f''$ as the function $f$ in the above formula,*\n",
    "\n",
    "$$\n",
    "I - T_1 = - \\int_a^b \\frac{f''(\\xi_x)}{2}(x-a)(b-x) \\, dx\n",
    "= - \\frac{f''(\\xi)}{2} \\int_a^b (x-a)(b-x) \\, dx\n",
    "$$\n",
    "\n",
    "*A bit of calculus gives $\\displaystyle \\int_a^b (x-a)(b-x) \\, dx = \\frac{(b-a)^3}{6}$, so*\n",
    "\n",
    "$$\n",
    "I - T_1  = -\\frac{f''(\\xi)}{2} \\frac{(b-a)^3}{6} = -\\frac{f''(\\xi)}{12} (b-a)^3 = -\\frac{f''(\\xi)}{12} h^3,\n",
    "$$\n",
    "\n",
    "*as advertised.*\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Proof of Theorem 2: the error in the Midpoint Rule\n",
    "\n",
    "*For this, we can use Taylor's Theorem for the linear approximation*\n",
    "\n",
    "$$f(x) = f(c) + f'(c)(x-c) + \\frac{f''(\\xi_x)}{2}(x-c)^2$$\n",
    "\n",
    "*with $c = (a+b)/2$, the midpoint. That is,*\n",
    "\n",
    "$$ f(x) - f(c) = f'(c) (x-c) + \\frac{f''(\\xi_x)}{2} (x-c)^2$$\n",
    "\n",
    "*and integrating each side gives*\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) -  f(c) \\, dx\n",
    "= I - M_1\n",
    "= \\int_a^b \\left[ f'(c)(x-c) + \\frac{f''(\\xi_x)}{2}(x-c)^2 \\right] dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Here symmetry helps, by eliminating the first (potentialy biggest) term in the error:\n",
    "we use the fact that $a = c - h/2$ and $b = c + h/2$*\n",
    "\n",
    "$$\n",
    "\\int_a^b f'(c)(x-c) \\, dx = f'(c) \\int_{c-h/2}^{c + h/2} x-c \\, dx = \\left[ (x-c)^2/2 \\right]_{c-h/2}^{c + h/2} = (h/2)^2 - (h/2)^2 = 0\n",
    "$$\n",
    "\n",
    "*Thus the error simplifies to*\n",
    "\n",
    "$$\n",
    "I - M_1 = \\int_a^b \\frac{f''(\\xi_x)}{2}(x-c)^2 \\, dx\n",
    "$$\n",
    "\n",
    "*and much as above, the Integral Mean Value Theorem can be used, this time with weight function $w(x) = (x-c)^2, \\geq 0$:*\n",
    "\n",
    "$$\n",
    "I - M_1 = \\frac{f''(\\xi)}{2} \\int_a^b (x-c)^2 \\, dx\n",
    "$$\n",
    "\n",
    "*Another caluclus exercise:\n",
    "$\\displaystyle \\int_a^b (x-c)^2 dx = \\int_{-h/2}^{h/2} x^2 dx = \\left[x^3/3\\right]_{-h/2}^{h/2} = h^3/12$,\n",
    "so indeed,*\n",
    "\n",
    "$$\n",
    "I - M_1 = \\frac{f''(\\xi)}{24} h^3\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"Left-hand-Rule\"></a>\n",
    "## Appendix: Approximating a Definite Integral With the Left-hand Endpoint Rule\n",
    "\n",
    "An even simpler approximation of $\\int_a^b f(x)\\, dx$ is the *Left-hand Endpoint Rule*,\n",
    "probably seen in a calculus course.\n",
    "For a single interval, this uses the approximation\n",
    "\n",
    "$$ f(x) \\approx f(a) $$\n",
    "\n",
    "leading to\n",
    "\n",
    "$$I := \\int_a^b f(x)\\, dx \\approx L_1 :=  \\int_a^b f(a) \\, dx = f(a) (b-a) $$\n",
    "\n",
    "The correpsonding composite rule with $n$ sub-intervals of equal width $h = (b-a)/n$ is\n",
    "\n",
    "$$L_n = \\sum_{i=0}^{n-1} f(x_i) h, \\; = \\sum_{i=0}^{n-1} f(a + i h) h$$\n",
    "\n",
    "with $x_i = a + i h$ as before."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 3.  Error in the Left-hand Endpoint Rule, $L_1$**\n",
    "\n",
    "For a function $f$ that is differentiable on interval $[a, b]$, the error in the Left-hand Endpoint Rule is\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) dx - L_1 = \\frac{(b-a)^2}{2}f'(\\xi), = \\frac{h^2}{2}f'(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proof:\n",
    "\n",
    "*This time use Taylor's Theorem just for the constant approximation wth center $a$:*\n",
    "\n",
    "$$ f(x) = f(c) + f'(\\xi_x)(x-a) $$\n",
    "\n",
    "That is,\n",
    "\n",
    "$$ f(x) - f(a) = f'(\\xi_x)(x-a)$$\n",
    "\n",
    "*so integrating each side gives*\n",
    "\n",
    "$$\n",
    "\\int_a^b f(x) -  f(a) \\, dx\n",
    "= I - L_1\n",
    "= \\int_a^b  f'(\\xi_x)(x-a) dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the Integral Mean Value Theorem again, now with weight $w(x) = x-a$ gives\n",
    "\n",
    "$$\n",
    "\\int_a^b f'(\\xi_x)(x-a) dx = f'(\\xi) \\int_a^b (x-a) dx = f'(\\xi) \\frac{(b-a)^2}{2} = \\frac{h^2}{2} f'(\\xi) \\; \\text{for some} \\; \\xi \\in [a, b]\n",
    "$$\n",
    "\n",
    "and inserting this into the previous formula gives the result."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "This work is licensed under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/)"
   ]
  }
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