// Numbas version: exam_results_page_options
{"name": "Find normals to surfaces enclosing a region, and the volume of the region", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b1"}, "theta2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi/2", "description": "", "name": "theta2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a1"}, "vol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(0.5*b1*(theta2-theta1)*a1^2,3)", "description": "", "name": "vol"}, "theta1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "theta1"}}, "ungrouped_variables": ["a1", "vol", "theta1", "b1", "theta2"], "name": "Find normals to surfaces enclosing a region, and the volume of the region", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Outward normals to the surfaces enclosing a region; volume of that enclosed region.</p>"}, "parts": [{"prompt": "<p>Find the unit outward normal to each component of the region's boundary.</p>\n        <ul>\n        <li>Unit outward normal at $x=0$: $($[[0]]$,$[[1]]$,$[[2]]$)$</li>\n        <li>Unit outward normal at $y=0$:&nbsp;$($[[3]]$,$[[4]]$,$[[5]]$)$</li>\n        <li>Unit outward normal at $z=0$:&nbsp;$($[[6]]$,$[[7]]$,$[[8]]$)$</li>\n        <li>Unit outward normal at $z=\\var{b1}$:&nbsp;$($[[9]]$,$[[10]]$,$[[11]]$)$</li>\n        <li>Unit outward normal at curved face:&nbsp;$($[[12]]$,$[[13]]$,$[[14]]$)$ (Enter your answer to 3d.p.)</li>\n        </ul>\n        <p>&nbsp;</p>", 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"statement": "<p>A region is enclosed by the surfaces $x^2+y^2\\leqslant\\var{a1^2}$, $0\\leqslant z\\leqslant\\var{b1}$, $x\\geqslant 0$, $y\\geqslant 0$.</p>", "tags": ["checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The unit outward normals can most easily be identified by sketching the region bounded by the given surfaces which, in this case, is a quarter cylinder.</p>\n    <p>Then the unit outward normals to the non-slanted surfaces are given by</p>\n    <ul>\n    <li>Unit outward normal at $x=0$: $\\pmatrix{-1,0,0}$</li>\n    <li>Unit outward normal at $y=0$:&nbsp;$\\pmatrix{0,-1,0}$</li>\n    <li>Unit outward normal at $z=0$:&nbsp;$\\pmatrix{0,0,-1}$</li>\n    <li>Unit outward normal at $z=\\var{b1}$:&nbsp;$\\pmatrix{0,0,1}$</li>\n    </ul>\n    <p>&nbsp;</p>\n    <p>The outward normal to the final, curved surface, is given by</p>\n    <p>\\[\\boldsymbol{\\nabla}(\\simplify{x^2+y^2-{a1^2}})=\\pmatrix{2x,2y,0},\\]</p>\n    <p>and so the unit outward normal is given by</p>\n    <p>\\[\\frac{1}{\\sqrt{x^2+y^2}}\\pmatrix{x,y,0}.\\]</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>The volume of a region $V$ bounded by some particular surfaces is given by</p>\n    <p>\\[V=\\int_V\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z,\\]</p>\n    <p>which in this case is</p>\n    <p>\\[V=\\int_R\\left[z\\right]_0^{\\var{b1}}\\mathrm{d}x\\mathrm{d}y,\\]</p>\n    <p>where $R$ is the projection of $V$ onto the $\\pmatrix{x,y}$-plane.</p>\n    <p>Due to the nature of the surface, however, it is convenient to simplify the integrals in $x$ and $y$, by using polar coordinates</p>\n    <p>\\[\\begin{align}x&amp;=r\\cos(\\theta),\\\\y&amp;=r\\sin(\\theta),\\end{align}\\]</p>\n    <p>then the integral becomes</p>\n    <p>\\[V=\\var{b1}\\int_{\\var{theta1}}^{\\frac{\\pi}{2}}\\mathrm{d}\\theta\\int_0^{\\var{a1}}r\\mathrm{d}r,\\]</p>\n    <p>which, after some straight forward integration, gives</p>\n    <p>\\[V=\\simplify{{b1*a1^2}/4}\\pi=\\var{vol}\\;\\text{to 3d.p.}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}