// Numbas version: exam_results_page_options
{"name": "Volume of revolution 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["vol", "m", "a1", "tol", "td1", "tvol", "c1", "d1"], "name": "Volume of revolution 2", "tags": ["Calculus", "calculus", "definite integration", "diagram", "integral", "integration", "rotation about an axis", "rotation about y-axis", "volume integral", "volume of revolution"], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>a)<br/>Given $y=\\var{a1}\\ln(\\var{m}x)$ we rearrange so that $x$ is the subject of the equation:<br/>\\[\\begin{eqnarray*} y&amp;=&amp;\\var{a1}\\ln(\\var{m}x)\\\\ \\Rightarrow \\ln(\\var{m}x) &amp;=&amp;\\frac{y}{\\var{a1}}\\\\ \\Rightarrow \\var{m}x&amp;=&amp; e^{\\frac{y}{\\var{a1}}}\\\\ \\Rightarrow x&amp;=&amp;\\frac{1}{\\var{m}}e^{\\frac{y}{\\var{a1}}} \\end{eqnarray*} \\]<br/>Hence \\[g(y)=\\frac{1}{\\var{m}}e^{\\frac{y}{\\var{a1}}}\\]</p>\n  <p>b)<br/>The volume of revolution is given by:<br/>\\[V=\\pi\\int_{\\var{c1}}^{\\var{d1}}g(y)^2\\;dy\\]<br/>Using the expression for $g(y)$ from the first part we have:<br/>\\[\\begin{eqnarray*} V&amp;=&amp;\\pi\\int_{\\var{c1}}^{\\var{d1}}\\frac{1}{\\var{m^2}}\\simplify[std]{e^(2y/{a1})}\\;dy\\\\ &amp;=&amp;\\simplify[std]{({a1}/{2*m^2})*pi}\\left[\\simplify[std]{e^(2y/{a1})}\\right]_{\\var{c1}}^{\\var{d1}}\\\\ \\\\&amp;=&amp;\\var{tvol}\\pi = \\var{vol}\\pi \\end{eqnarray*} \\]<br/>Hence the multiple of $\\pi$ to two decimal places is $\\var{vol}$.</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>Rewrite this function in the form $x=g(y)$ , where $g(y)$ is a function of $y$ only.</p>\n      \n      \n      \n      <p>$x=g(y)=\\;\\;$[[0]]</p>\n      \n      \n      ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 / {m}) * Exp(y / {a1})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>Hence, find the volume of revolution, $V$ obtained as follows:</p>\n      <p>Rotate $y=f(x)$ by $2\\pi$ radians about the $y$-axis, between the limits of $y=\\var{c1}$ and $y=\\var{d1}$.</p>\n      <p>The volume $V$ can be written as a multiple of $\\pi$, $V=m\\pi$, where:</p>\n      <p>$m=\\;\\;$[[0]]. &nbsp;Input $m$ to two decimal places.</p>\n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "vol+tol", "minValue": "vol-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Consider the function \\[y=f(x)=\\var{a1}\\ln(\\var{m}x)\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"vol": {"definition": "precround(tvol,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "vol", "description": ""}, "m": {"definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "a1": {"definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "td1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "td1", "description": ""}, "tvol": {"definition": "((a1)/(2*m^2))*(exp(2*d1/a1)-exp(2*c1/a1))", "templateType": "anything", "group": "Ungrouped variables", "name": "tvol", "description": ""}, "c1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "d1": {"definition": "if(td1<=c1,c1+1,td1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>3/07/2012:</strong></p>\n    \t\t    <p>Added tags.</p>\n    \t\t    <p>Improved display in statement and prompts, and Advice.</p>\n    \t\t    <p>No tolerance allowed in second part answer. Set new tolerance variable tol=0.</p>\n    \t\t    <p>Checked calculation.</p>\n    \t\t    <p><strong>20/07/2012:</strong></p>\n    \t\t    <p>Added description.</p>\n    \t\t    <p>Changed question format to hopefully make it clear that it is the multiple of $\\pi$ wanted.</p>\n    \t\t    <p>Checked calculation again.</p>\n    \t\t    <p>Should have a diagram of the volume - or a schematic version of revolving a function about the $y$-axis.</p>\n    \t\t    <p></p>\n    \t\t    <p><strong>25/07/2012:</strong></p>\n    \t\t    <p>&nbsp;</p>\n    \t\t    <p>Added tags.</p>\n    \t\t    <p></p>\n    \t\t    <p>In the Advice section moved \\Rightarrow so that it is at the beginning of the line instead of the end of the previous line.</p>\n    \t\t    <p>&nbsp;</p>\n    \t\t    <p>&nbsp;</p>\n    \t\t    <p>Question appears to be working correctly.</p>\n    \t\t    <p>&nbsp;</p>\n    \t\t", "description": "<p>Rotate the graph of &nbsp;$y=a\\ln(bx)$&nbsp;&nbsp;by $2\\pi$ radians about the $y$-axis between $y=c$ and $y=d$. Find the volume of revolution.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}