// Numbas version: finer_feedback_settings
{"name": "Volume of revolution 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "tv", "v", "sb", "sa"], "name": "Volume of revolution 1", "tags": ["Calculus", "calculus", "definite integration", "diagram", "integral", "integration", "rotation about an axis", "rotation about x axis", "volume integral", "volume of revolution"], "preamble": {"css": "", "js": ""}, "advice": "<p>Recall that if $V$ is the volume generated between the limits $x=a$ and $x=b$ by rotating the function about the $x$-axis then $\\displaystyle V=\\pi\\int_a^by^2\\;dx$.</p>\n<p>So we have:<br/>\\[\\begin{eqnarray*} V&amp;=&amp;\\pi\\int_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\simplify[std]{{a^2}(cos(x)+{b})^2}\\;dx\\\\ &amp;=&amp;\\var{a^2}\\pi\\int_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\simplify[std]{cos(x)^2+{2*b}*cos(x)+{b^2}}\\;dx\\\\ &amp;=&amp;\\var{a^2}\\pi\\left[\\simplify[std]{((1 / 4) Sin(2*x) + (1 / 2) * x + {2 * b} * Sin(x) + {b ^ 2} * x)}\\right]_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\\\ \\end{eqnarray*}\\]<br/>Here we have used the identity $\\cos(x)^2=\\frac{1}{2}(1+\\cos(2x))$ in order to integrate $\\cos(x)^2$.</p>\n<p>Since $\\sin(n\\pi)=0$ for all integers $n$ we see that:<br/>\\[\\begin{eqnarray*} V&amp;=&amp;\\var{a^2}\\pi\\frac{\\var{1+2b^2}}{\\var{2}}\\left(\\var{c+2}\\pi-\\var{c}\\pi\\right)\\\\ &amp;=&amp;\\var{a^2*(1+2b^2)}\\pi^2\\\\ &amp;=&amp;\\var{V}\\mbox{ to 3 decimal places} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "\n      <p>Find the volume of this object.</p>\n        <p>$V=\\;\\;$[[0]]</p>\n        <p>Enter your answer to 3 decimal places.</p>\n        <p>Click on Show steps for information on volumes of revolution. You will not lose any marks.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>Recall that if $V$ is the volume generated between the limits $x=a$ and $x=b$ &nbsp;then $\\displaystyle V=\\pi\\int_a^by^2\\;dx$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "V", "minValue": "V", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Consider the solid object that is obtained when the function: \\[y=\\simplify[std]{{a}(cos(x)+{b})}\\] is rotated by $2\\pi$ radians about the $x$-axis between the limits $x=\\var{c}\\pi$ and $x=\\var{c+2}\\pi$</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "sa*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "sb*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "tv": {"definition": "pi^2*a^2*(1+2*b^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "tv", "description": ""}, "v": {"definition": "precround(tV,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "sb": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "sb", "description": ""}, "sa": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "sa", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>3/07/2012:</strong></p>\n    \t\t    <p>Added tags.</p>\n    \t\t    <p>Checked calculations.</p>\n    \t\t    <p>Improved display of statement, prompt and Advice.&nbsp;</p>\n    \t\t    <p>Wanted to put the Hint into Show steps - but cannot create Steps at present.</p>\n    \t\t    <p>No tolerance allowed. Must be exact to three decimal places.</p>\n    \t\t    <p>Note the use of $\\cos(x)^2$ instead of the standard $\\cos^2(x)$ as best to be consistent as we cannot use $\\cos^2(x)$ if any jme calculation is involved.</p>\n    \t\t    <p><strong>20/07/2012:</strong></p>\n    \t\t    <p>Added description.</p>\n    \t\t    <p>Added Show steps hint.</p>\n    \t\t    <p>Checked description.</p>\n    \t\t    <p>Perhaps the tolerance should be 1, not 0.001 given the magnitude of the answer.</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>&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 $y=a(\\cos(x)+b)$ by $2\\pi$ radians about the $x$-axis between $x=c\\pi$ and $x=(c+2)\\pi$. 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/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}