{
  "cells": [
     {"cell_type":"markdown","source":"<h1>Questions to be handed in for Applications of Integration</h1>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>Begin by loading our package for plotting, our root-finding package, and symbolic math package:</p>","metadata":{}},
{"outputs":[],"cell_type":"code","source":["using Plots\nusing Roots\nusing SymPy\nusing QuadGK"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>We discuss a few applications of the integral.</p>","metadata":{}},
{"cell_type":"markdown","source":"<h2>Area between two functions</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<blockquote>\n<p>The area between the two functions $f(x) > g(x)$ between $a$ and $b$</p>\n</blockquote>","metadata":{}},
{"cell_type":"markdown","source":"<p>is given by $\\int_a^b (f(x) - g(x)) dx$.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p><a href=\"https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/definite_integrals/v/area-between-curves\">Kahn academy</a></p>","metadata":{}},
{"cell_type":"markdown","source":"<p>For example, let $f(x) = e^{-x}$ and $g(x) = x^2$. Find the area below $f$ and above $g(x)$ between $0$ and the first positive intersection point.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = exp(-x)\ng(x) = x^2\nh(x) = f(x) - g(x)\nip = find_zero(h, 1)\nplot(f, 0, ip)\nplot!(g)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>We can then find the area using <code>quadgk</code> to perform the (numeric) integration:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0.3890927851164812, 0.0)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["quadgk(h, 0, ip)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>If you don't care about performance, you can avoid finding the zero through a trick like this:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0.38909278887340365, 1.6320640461450628e-9)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["h(x) = f(x) - g(x) > 0 ? f(x) - g(x) : 0.0\nquadgk(h, 0, 1)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Just choose the endpoints of the integration so that you don't pick up extra area. In this case, you can easily plot to see that they don't cross again after <code>ip</code>, so any number greater will work.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>(The performance is killed, because <code>quadgk</code> can possibly perform many function evaluations. When you put in a condition to check each time it can take much more time. In this example, though it isn't too much extra work, as the functions are well behaved.)</p>","metadata":{}},
{"cell_type":"markdown","source":"<h2>Volume of solids of revolution</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<blockquote>\n<p>The volume of the [solid of</p>\n</blockquote>","metadata":{}},
{"cell_type":"markdown","source":"<p>revolution](http://en.wikipedia.org/wiki/Solid_of_revolution) found   by rotating the graph of $f(x)$ around the $x$ axis between $a$ and   $b$ is given by</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nVolume = \\int_a^b \\pi f(x)^2 dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>For example, rotating the line $y=1+ x/10$ around the $x$ axis between $0$ and $5$ will produce a <a href=\"http://en.wikipedia.org/wiki/Frustum\">frustum</a> of a cone. It has volume is given by:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(24.870941840919194, 0.0)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(x) = 1 + x/10\nv(x) = pi * r(x)^2\nquadgk(v, 0, 5)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h2>Volumes with known cross-sections</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>The above is a special case of the following:</p>","metadata":{}},
{"cell_type":"markdown","source":"<blockquote>\n<p>If $c(x)$ gives the cross sectional area of a volume at height $x$,</p>\n</blockquote>","metadata":{}},
{"cell_type":"markdown","source":"<p>then the total volume from $a$ to $b$ is given by </p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nVolume = \\int_a^b c(x) dx.\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>For example, if we let $y=1 + x/10$, again, but this time form squares with half distance given by $y$, the total volume becomes:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(31.66666666666667, 0.0)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["r(x) = 1 + x/10\nc(x) = (2r(x))^2\t\t# area of square is d * d with d= 2r\nquadgk(c, 0, 5)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>This is a volume of a frustum of a pyramid with 4 sides, of which the previous partial cone would just fit inside. As such, this latter volume, 31.67, is greater than the previous. Even without this view, you would know this as the cross sectional squares are larger in area than the cross sectional circles formed by revolution. That is $\\pi \\cdot r(x)^2 < (2r(x))^2$.</p>","metadata":{}},
{"cell_type":"markdown","source":"<h3>Questions</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<h4>Area between two curves</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Find the amount of area between $f(x) = x^3$ and $g(x) = x^4$ between $0$ and $1$.</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Area between two curves</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Find the area between the curves $\\cos(x)$ and $\\sin(x)$ between $-\\pi$ and $\\pi$ <strong>when</strong> $\\cos(x) \\geq \\sin(x)$.</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>I like it – but how much does it weigh?</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Look at the sculpture [Le</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>Tamanoir](http://www.sculptureinternationalrotterdam.nl/collectie/permanent/LeTamanoir.php?lang=en) by Calder. A large scale work. How much does it weigh? Approximately?</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>Let's try to answer that with an educated guess. The right most figure looks to be about 1/5th the total amount. So if we estimate that piece and multiply by 5 we get a good guess. That part looks like an area of metal bounded by two quadratic polynomials. If we compute that area in square inches, then multiply by an assumed thickness of one inch, we have the cubic volume. The density of galvanized steel is 7850 kg/$m^3$ which we convert into pounds/in$^3$ via:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["0.28452123585283234"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["7850 * 2.2 * (1/39.3)^3"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>The two parabolas, after rotating, might look like the following (with $x$ in inches):</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nf(x) = x^2/70 \\quad g(x) = 35 + x^2/140\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>Put this altogether to give an estimated weight in pounds. Would you guess that the entire sculpture is more or less than a ton? two tons?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Archimedes and areas related to parabolas</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>For any quadratic polynomial, Archimedes proved that the the area bounded by the secant line between $[a,b]$ is known multiple more than the area of the triangle constructed by connecting the points on the parabola at $x=a$, $x=b$ and $x=(a + b)/2$.  <a href=\"http://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola#mediaviewer/File:Parabola_and_inscribed_triangle.svg\">Figure</a></p>\n<p>We want to find that multiple. Since it doesn't depend on the parabola, $a$ or $b$, take $f(x) = x^2$, $a=-2$ and $b=1$.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>Here is how we can compute the area bounded by the secant line between $a$ and $b$ and the parabola:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(4.5, 8.881784197001252e-16)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["f(x) = x^2\na,b = -2, 1\nmid = (a + b)/2\nsecant(f, a, b) = x->f(a) + (f(b)-f(a))/(b-a)*(x-a)\ntopline = secant(f,a,b)\nh(x) = topline(x) - f(x)\na, err = quadgk(h, a, b)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Now compute the other area of the triangle and take a ratio to find the answer.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Area between two curves</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Center a square of width 2 at the origin. Find the area of all points $(x,y)$ where the distance to the origin is less than the distance to the boundary of the square.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>The symmetry of a graph will show that it might pay to focus on the region given by $y > |x|$ for which the closest edge is the top, so for a point $(x,y)$ in that region, the curve satisfies $1 - y = \\sqrt{x^2 + y^2}$ which you can solve symbolically through:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{bmatrix}- \\frac{x^{2}}{2} + \\frac{1}{2}\\end{bmatrix}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["@vars x y\nsolve(1-y-sqrt(x^2+y^2), y)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>What is the total area? Is it more than half the total area?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Volumes <small>rotated catenary</small></h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $f(x; a) = a \\cdot \\cosh(x/a)$. For $a=0.5$ find the volume of the figure of revolution if the graph of $f$ is rotated around the $x$ axis between $-a$ and $a$.</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Volume of a watermelon</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>A watermelon is modeled as an ellipsoid – the solid formed by rotating an ellipse. Suppose the dimensions of a particular watermelon has a major axis of 30cm and a minor axis of 20 cm. What is the volume in cubic centimeters? What is the weight if 1000 cubic centimeters weighs $(2/3) \\cdot 2.2$ pounds (about 2/3rd the weight of plain water).</p>\n<p>(You might model the watermelon via $(x/15)^2  + (y/10)^2 = 1$ and rotate about the $x$ axis..)</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Volume of a dome</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>The Onion Domes of <a href=\"http://en.wikipedia.org/wiki/Saint_Basil&#39;s_Cathedral\">Saint Basil's Cathedral</a> are iconic figures.  There are theories as to the origins of the shape, such as they don't hold snow and they give more mass to support the large crosses on top of them.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>These domes can be viewed as solids of revolution.  We model the relationship using functions presented <a href=\"http://math.clarku.edu/~djoyce/ma131/surfaces.pdf\">here</a>, only scaled to more closely match reality.  The functional form presented is equivalent to</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\nr(y) = A \\cdot (1 + \\cos(y/B)),\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>where the radius $r(0)$ is equal to 20 and the height satisfies $(\\pi+ \\pi/3) \\cdot B = 40$. The dome is modeled by the solid of revolution formed by rotating the above curve around the $y$ axis from $-10$ to $30$. If the units are in feet, what is the volume of the dome?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Volumes <small>Washington Monument</small></h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>The <a href=\"http://en.wikipedia.org/wiki/Washington_Monument\">Washington Monument</a> is the world's tallest stone structure.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>The basic shape is the 500 foot high base consists of square cross sections with diameter tapering down linearly from 126 feet to 40 feet. In the last 55 feet it is a square-based pyramid.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>The cross section as a function of height, $x$, can be written in terms of the following function, $f$ as $c(x) = (2f(x))^2$.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["f (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["f(x) = x < 500 ? 126.0 + (40-126)/(500-0)*x : 40.0 + (0 - 40)/(55-0)*(x-500)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Find the total volume in cubic feet.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>Volumes <small>rotating a trapped area</small></h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>A generalization of solid of revolution consists of rotating a region bounded between $g(x) < f(x)$ around the $x$ asis. Such a solid with have volume $V =\\int_a^b \\pi(f(x)^2 - g(x)^2) dx$, as might be guessed.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>A ball of radius $1$ has a core taken out of it with radius $1/2$. Find the remaining volume. (This is a solid of revolution for the functions $f(x) = \\sqrt{1-x^2}$ and $g(x) = 1/2$, but the $a$ and $b$ need determining.)</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h4>An AP calculus exam question</h4>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $R$ be the region enclosed by the graph $f(x) = x^4 - 2.3x^3 + 4$ and the horizontal line $y=4$, as shown by</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = x^4 - 2.3x^3 + 4\ng(x) = 4\nplot(f, 0, 2.5)\nplot!(g)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ol>\n<li><p>Find the volume of the solid generated when $R$ is rotated about</p>\n</li>\n</ol>","metadata":{}},
{"cell_type":"markdown","source":"<p>the horizontal line $y=-2$.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ol start=\"2\">\n<li><p>Region $R$ is the base of a solid. For this solid, each cross</p>\n</li>\n</ol>","metadata":{}},
{"cell_type":"markdown","source":"<p>section perpendicular to the $x$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}
    ],
 "metadata": {
  "language_info": {
   "file_extension": ".jl",                                                                                                             
   "mimetype": "application/julia", 
   "name": "julia",
   "version": "0.6"
  },
 "kernelspec": {
   "display_name": "Julia 0.6.0",
   "language": "julia",
   "name": "julia-0.6"
  }

 },
 "nbformat": 4,
 "nbformat_minor": 2

}