{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Definite Integrals, Part 1: The Building Blocks\n",
    "\n",
    "**References:**\n",
    "\n",
    "- Sections 5.2.1, 5.2.3 and 5.2.4 of [Sauer](../references.html#Sauer)\n",
    "\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 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": 1,
   "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": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 1.0\n",
    "b = 3.0\n",
    "f(x) = exp(x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "Figure(PyObject <Figure size 1200x600 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=[12, 6])\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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 1200x600 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_midpoint = f((a+b)/2)\n",
    "figure(figsize=[12, 6])\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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 1200x600 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f_midpoint = f((a+b)/2)\n",
    "figure(figsize=[12, 6])\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": [
    "**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",
    "{doc}`polynomial-collocation-error-formulas-julia`,\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",
    "This work is licensed under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/)"
   ]
  }
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