06-double_triple_integrals_project.ipynb

{
  "cells": [
     {"cell_type":"markdown","source":"<h1>Questions to be handed in on using Julia for multiple integration</h1>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>(Thanks to Thomas' Calculus for the problems)</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>These questions use <code>SymPy</code> for symbolic manipulations and <code>Plots</code> for graphing.</p>","metadata":{}},
{"outputs":[],"cell_type":"code","source":["using Plots, SymPy"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h2>Questions</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Use <code>SymPy</code> to compute</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int_0^1 \\int_0^2 (4 - y^2) dy dx\n$$\n","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":"<ul>\n<li><p>Integrate</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int_1^2 \\int_y^{y^2} dx dy.\n$$\n","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":"<ul>\n<li><p>Integrate $f(x,y) = x/y$ over the region in the first quadrant bounded by the lines $y=x$, $y=2x$, $x=1$ and $x=2$.</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":"<ul>\n<li><p>Integrate $f(u,v) = v - \\sqrt{u}$ over the triangular region in the first quadrant of the $uv$-plane cut by the line $u + v = 1$.</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":"<ul>\n<li><p>Integrate the following</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int_0^\\pi \\int_x^\\pi \\frac{\\sin(y)}{y} dy dx\n$$\n","metadata":{}},
{"cell_type":"markdown","source":"<p>by first switching the limits of integration and then integrating.</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":"<h2>Applications related to the center of mass</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Find the center of mass of a thin plate of density $\\delta = 3$ located in the first quadrant and bounded by the lines $x=0$, $y=x$ and the parabola $y=2-x^2$.</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":"<ul>\n<li><p>It can cause quite a stir when an appliance – like a vending machine – tips over. Suppose a manufacturer makes a parabolic shaped appliance with profile $y = a(1-x^2)$. What values of $a$ will ensure that if the appliance is tipped no more than 45 degrees it will not tip? (The center of mass should be below the line $y = 1 - x$.)</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":"<h2>Double integrals in polar form</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Change the Cartesian integral into an equivalent polar integral and evaluate that:</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int_{-1}^{1} \\int_{-\\sqrt{1-y^2}}^{0} \\frac{4 \\sqrt{x^2 + y^2}}{1 + x^2 + y^2} dx dy.\n$$\n","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":"<ul>\n<li><p>Find the area enclosed by one leaf of the rose $r=12 \\cos(3\\theta)$.</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":"<ul>\n<li><p>Find the area of the region common to the interiors of the cardioids $r=1+\\cos(\\theta)$ and $r=1-\\cos(\\theta)$. A plot gives an indication on how one may proceed:</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":["using SymPy, Plots\n\nts = range(0, stop=2pi, length=100)\nt1s = range(pi + pi/4, stop=2pi, length=100)\nt2s = range(pi/2, stop=pi + pi/4, length=100)\nr1(t) = 1 - cos(t)\nr2(t) = 1 - sin(t)\nx1s = [r1(t) * cos(t) for t in t1s]\ny1s = [r1(t) * sin(t) for t in t1s]\nx2s = [r2(t) * cos(t) for t in t2s]\ny2s = [r2(t) * sin(t) for t in t2s]\n\nplot(x1s, y1s)\nplot!(x2s, y2s)"],"metadata":{},"execution_count":1},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h3>Triple integrals</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Evaluate the integral</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\int_0^1 \\int_0^{2-x} \\int_0^{2 - x - y} dz dy dx.\n$$\n","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":"<ul>\n<li><p>Find the volume of the region $D$ enclosed by the surfaces $z=x^2 + 3y^2$ and $z=8-x^2 - y^2$.</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}
    ],
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   "file_extension": ".jl",
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   "version": "0.6"
  },
 "kernelspec": {
   "display_name": "Julia 1.0.0",
   "language": "julia",
   "name": "julia-1.0"
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 },
 "nbformat": 4,
 "nbformat_minor": 2

}