// Numbas version: exam_results_page_options
{"showQuestionGroupNames": false, "duration": 0, "metadata": {"description": "<p>5 questions on definite integrals - integrate polynomials, trig functions and exponentials; find the area under a graph; find volumes of revolution.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "navigation": {"browse": true, "onleave": {"message": "", "action": "none"}, "allowregen": true, "preventleave": false, "showresultspage": "oncompletion", "reverse": true, "showfrontpage": false}, "name": "mathcentre: Definite integration ", "feedback": {"showtotalmark": true, "showanswerstate": true, "showactualmark": true, "advicethreshold": 0, "intro": "", "allowrevealanswer": true, "feedbackmessages": []}, "showstudentname": true, "timing": {"timeout": {"message": "", "action": "none"}, "timedwarning": {"message": "", "action": "none"}, "allowPause": true}, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 1, "name": "Group", "questions": [{"name": "Definite integration 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n  <p>a)<br/>\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\] <br/>Expand the parentheses to obtain:</p>\n    <p>\\[\\begin{eqnarray*}I &amp;=&amp; \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &amp;=&amp;\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &amp;=&amp;\\var{tans1}\\\\ \\\\&amp;=&amp;\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>c)<br/>\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\] <br/>We use integration by parts.</p>\n    <p>Recall that:<br/>\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]<br/>Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$</p>\n    <p>Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]</p>\n    <p>So \\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &amp;=&amp;\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &amp;=&amp; \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]<br/>d)</p>\n    <p>\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]</p>\n    <p>Use integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$<br/>\\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &amp;=&amp;\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &amp;=&amp;\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \\]</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 2 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 2 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 2 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 4 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans4-tol1", "type": "numberentry", "maxvalue": "ans4+tol1", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Evaluate the following definite integrals.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"ans1": {"definition": "precround(tans1,2)", "name": "ans1"}, "ans2": {"definition": "precround(ln(1+b2/m2),2)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,2)", "name": "ans3"}, "ans4": {"definition": "precround(tans4,4)", "name": "ans4"}, "b4": {"definition": "s7*random(1,2,3)", "name": "b4"}, "b1": {"definition": "random(2..6)", "name": "b1"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "d1": {"definition": "random(-9..9)", "name": "d1"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s7": {"definition": 1.0, "name": "s7"}, "s6": {"definition": -1.0, "name": "s6"}, "m4": {"definition": 2.0, "name": "m4"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "tol": {"definition": 0.01, "name": "tol"}, "a1": {"definition": "random(1..7)", "name": "a1"}, "tans4": {"definition": "(e^(p)*(p^2-2*p+2)-2)/(n4^3)", "name": "tans4"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "a1^2*(b1^5-1)/5+a1*c1*(b1^4-1)/2+(2*a1*d1+c1^2)*(b1^3-1)/3+c1*d1*(b1^2-1)+d1^2*(b1-1)", "name": "tans1"}, "tans3": {"definition": "if(w=0,((-1)^(m3)-1)/m3^2,-pi*(-1)^(m3)/m3)", "name": "tans3"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "p": {"definition": "n4*b4", "name": "p"}, "t": {"definition": "random(1,-1)", "name": "t"}, "w": {"definition": "random(0,1)", "name": "w"}, "n4": {"definition": "s6*random(1,2,3)", "name": "n4"}}, "metadata": {"notes": "\n    \t\t<p><strong>3/07/1012:</strong></p>\n    \t\t        <p>Added tags.</p>\n    \t\t        <p>Checked calculations.</p>\n    \t\t        <p>Left tolerances in, as easy to make minor errors in calculations.</p>\n    \t\t        <p>Improved display in Advice.</p>\n    \t\t        <p>Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.</p>\n    \t\t        <p><strong>20/07/2012:</strong></p>\n    \t\t        <p>Set new tolerace variables, tol=0.01, tol1=0.0001.</p>\n    \t\t        <p>Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.</p>\n    \t\t        <p>Added description.</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>A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?</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        <p>&nbsp;</p>\n    \t\t", "description": "<p>Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Definite Integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n  <p>&nbsp;</p>\n    <p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>\\[I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]</p>\n      <p>$I=\\;\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Evaluate the following definite integrals.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-2..2#0.5 except 0)", "name": "a"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "b": {"definition": "random(2..5)", "name": "b"}, "w": {"definition": "random(0,1)", "name": "w"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "ans2": {"definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,3)", "name": "ans3"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "b1": {"definition": "random(-1..2#0.5 except 0)", "name": "b1"}, "tol": {"definition": 0.001, "name": "tol"}, "t": {"definition": "random(1,-1)", "name": "t"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "tans3": {"definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3"}, "d1": {"definition": "random(-9..9)", "name": "d1"}}, "metadata": {"notes": "\n    \t\t                \t\t<p><strong>3/07/1012:</strong></p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>Checked calculations.</p>\n    \t\t                \t\t        <p>Left tolerances in, as easy to make minor errors in calculations.</p>\n    \t\t                \t\t        <p>Improved display in Advice.</p>\n    \t\t                \t\t        <p>Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.</p>\n    \t\t                \t\t        <p><strong>20/07/2012:</strong></p>\n    \t\t                \t\t        <p>Set new tolerace variables, tol=0.01, tol1=0.0001.</p>\n    \t\t                \t\t        <p>Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.</p>\n    \t\t                \t\t        <p>Added description.</p>\n    \t\t                \t\t        <p></p>\n    \t\t                \t\t        <p><strong>25/07/2012:</strong></p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \n    \t\t", "description": "<p>Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["c", "b", "b2", "ans1", "ans2", "t", "tol", "ans", "b1"], "tags": ["areas", "definite integration", "integration"], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>First we observe that:\\[\\simplify[std]{int ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x)={t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))+C}.\\]</p>\n    <p>Hence we have:</p>\n    <p>\\[\\begin{eqnarray*} \\simplify[std]{defint ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x,{b1},{b2})}&amp;=&amp;\\left[\\simplify[std]{{t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))}\\right]_{\\var{b1}}^{\\var{b2}}\\\\&amp;=&amp;\\var{ans}\\end{eqnarray*}\\]</p>\n    <p>to 3 decimal places.</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Enter the area $A$ here to 3 decimal places:</p>", "allowFractions": false, "marks": "1", "maxValue": "ans+tol", "minValue": "ans-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Find the area $A$ of the shape bounded by the $x$-axis, the function $y=\\simplify[std]{{t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c})}$ and the lines $x=\\var{b1},\\;x=\\var{b2}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"c": {"definition": "random(3..7 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "ans1": {"definition": "precround(c*(exp(b*b2/c)-exp(b*b1/c))/b,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround(1/b*((b*b2+c)*(ln(b*b2+c)-1)-(b*b1+c)*(ln(b*b1+c)-1)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "t": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "b2": {"definition": "b1+random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "ans": {"definition": "t*ans1+(1-t)*ans2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"notes": "<p>Changed the variables from the original Maple TA Leicester examples so that realistic answers obtained.</p>", "description": "<p>Finding areas under graphs using definite integration.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Volume of revolution 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "tv", "v", "sb", "sa"], "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": []}]}, {"name": "Volume of revolution 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["vol", "m", "a1", "tol", "td1", "tvol", "c1", "d1"], "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": []}]}]}], "percentPass": 0, "type": "exam", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}], "extensions": [], "custom_part_types": [], "resources": []}