{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Definite Integrals, Part 1: The Building Blocks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Sections 5.2.1 and 5.2.4 of Chapter 5 *Numerical Differentiation and Integration* in {cite}`Sauer`.\n",
    "- Sections 4.3 *Elmenets of Numerical Integration* of {cite}`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 particular, 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; the former will be visited soon."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot"
   ]
  },
  {
   "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": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 1.0\n",
    "b = 3.0\n",
    "f(x) = exp(x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x500 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_average = (f(a) + f(b))/2\n",
    "x = range(a, b, 100)\n",
    "figure(figsize=[10,5])\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_average, f_average, 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": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x500 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_midpoint = f((a+b)/2)\n",
    "figure(figsize=[10,5])\n",
    "plot(x, f.(x))\n",
    "plot([a, a, b, b, a], [0, f_midpoint, f_midpoint, 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": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 1000x500 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_midpoint = f((a+b)/2)\n",
    "figure(figsize=[10,5])\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_average, f_average, 0, 0], \"-.\", label=\"Trapezoid Rule area\")\n",
    "plot([a, a, b, b, a], [0, f_midpoint, f_midpoint, 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": [
    "```{prf:theorem} Error in the Trapezoid Rule, $T_1$\n",
    ":label: trapezoid-rule-error\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",
    "$$ \\int_a^b f(x) dx - T_1 = -\\frac{h^3}{12}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b]. $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} Error in the Midpoint Rule, $M_1$\n",
    ":label: midpoint-rule-error\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",
    "$$ \\int_a^b f(x) dx - M_1 = \\frac{h^3}{24}f''(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b] $$\n",
    "```"
   ]
  },
  {
   "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": [
    "To help prove the above formulas, we introduce a result that also helps in various places later:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:theorem} The Integral Mean Value Theorem\n",
    ":label: 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",
    "$$ \\int_a^b f(x) w(x) \\, dx = \\int_a^b f(\\xi) w(x) \\, dx, = f(\\xi) \\int_a^b w(x) \\, dx $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf: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",
    "$$ L w(x) \\leq f(x) w(x) \\leq H w(x) $$\n",
    "\n",
    "and by integrating,\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",
    "Dividing by $\\int_a^b w(x) \\,dx$ (which is positive),\n",
    "\n",
    "$$ L \\leq \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} \\leq H $$\n",
    "\n",
    "and the Mean Value Theorem says that $f$ attains this value for some $\\xi \\in [L, H]$:\n",
    "\n",
    "$$ f(\\xi) = \\frac{\\int_a^b f(x) w(x) \\,dx}{\\int_a^b w(x) \\,dx} $$ (integral-mean-value-formula)\n",
    "\n",
    "Clearing the denominator gives the claimed result.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "(of {prf:ref}`trapezoid-rule-error`, {prf:ref}`the trapezoid rule error formula) <trapezoid-rule-error>`\n",
    "\n",
    "The function integrated to get the Trapezoid Rule is the linear collocating polynomial $L(x)$,\n",
    "and from the section {doc}`polynomial-collocation-error-formulas`, 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",
    "$$\n",
    "\\int_a^b (f(x) - L(x))\n",
    "= I - T_1\n",
    "= \\int_a^b \\frac{f''(\\xi_x)}{2}(x-a)(x-b) \\, dx\n",
    "$$\n",
    "\n",
    "To get around the complication that $\\xi_x$ depends on $x$ in an unknown way, use the Ingtgral Measn Vlaue Theorem with weight function\n",
    "$w(x) = (x-a)(b-x), \\geq 0$ for $a \\leq x \\leq b$.\n",
    "Then with $-f''$ as the function $f$ in {eq}`integral-mean-value-formula`:\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",
    "A bit of calculus gives $\\displaystyle \\int_a^b (x-a)(b-x) \\, dx = \\frac{(b-a)^3}{6}$, so\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",
    "as advertised.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "(of {prf:ref}`midpoint-rule-error`, {prf:ref}`the midpoint rule error formula) <midpoint-rule-error>`\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",
    "$$\n",
    "\\int_a^b f(x) -  f(c) \\, dx\n",
    "= I - M_1\n",
    "= \\int_a^b \\left[ f'(c)(x-c) + \\frac{f''(\\xi_x)}{2}(x-c)^2 \\right] dx\n",
    "$$\n",
    "\n",
    "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",
    "$$ \\int_a^b f'(c)(x-c) \\, dx = f'(c) \\int_{c-h/2}^{c + h/2} x-c \\, dx = \\left[ (x-c)^2/2 \\right]_{c-h/2}^{c + h/2} = (h/2)^2 - (h/2)^2 = 0 $$\n",
    "\n",
    "Thus the error simplifies to\n",
    "\n",
    "$$ I - M_1 = \\int_a^b \\frac{f''(\\xi_x)}{2}(x-c)^2 \\, dx $$\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",
    "$$ I - M_1 = \\frac{f''(\\xi)}{2} \\int_a^b (x-c)^2 \\, dx $$\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",
    "$$ I - M_1 = \\frac{f''(\\xi)}{24} h^3 $$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(left-endpoint-rule)=\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": [
    "```{prf:theorem} Error in the Left-hand Endpoint Rule, $L_1$\n",
    ":label: error-left-endpoint-rule\n",
    "\n",
    "For a function $f$ that is differentiable on interval $[a, b]$, the error in the Left-hand Endpoint Rule is\n",
    "\n",
    "$$ \\int_a^b f(x) dx - L_1 = \\frac{(b-a)^2}{2}f'(\\xi), = \\frac{h^2}{2}f'(\\xi) \\quad \\text{for some} \\; \\xi \\in [a, b]n$$\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:proof}\n",
    "\n",
    "This time use Taylor's Theorem just for the constant approximation with 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",
    "$$\n",
    "\\int_a^b f(x) -  f(a) \\, dx\n",
    "= I - L_1\n",
    "= \\int_a^b  f'(\\xi_x)(x-a) dx\n",
    "$$\n",
    "\n",
    "Using the Integral Mean Value Theorem again, now with weight $w(x) = x-a$ gives\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",
    "and inserting this into the previous formula gives the result.\n",
    "```"
   ]
  }
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