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{"name": "1. Simple Polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["a", "b", "a2", "b2", "c", "d", "k", "n", "m"], "name": "Original variables"}, {"variables": ["k1", "k2", "k3", "k4", "k5"], "name": "Combined variables"}], "functions": {}, "preamble": {"js": "", "css": ""}, "name": "1. Simple Polynomial", "metadata": {"licence": "None specified", "description": "<p>I created this question, and every other question in Multiple Integration, for my dissertation `Computer-Aided Assessment of Multiple Integration'.</p>"}, "variables": {"k4": {"group": "Combined variables", "templateType": "anything", "definition": "k3/(m+1)", "description": "", "name": "k4"}, "m": {"group": "Original variables", "templateType": "anything", "definition": "random(1..2 except n)", "description": "<p>power of y</p>", "name": "m"}, "k1": {"group": "Combined variables", "templateType": "anything", "definition": "k/(n+1)", "description": "", "name": "k1"}, "b": {"group": "Original variables", "templateType": "anything", "definition": "random(0..(3-b2))", "description": "<p>start of y range&nbsp;{-5-c,5-c}</p>", "name": "b"}, "k2": {"group": "Combined variables", "templateType": "anything", "definition": "(c)^(n+1)-a^(n+1)", "description": "", "name": "k2"}, "a2": {"group": "Original variables", "templateType": "randrange", "definition": "random(1..3#1)", "description": "<p>x range</p>", "name": "a2"}, "n": {"group": "Original variables", "templateType": "randrange", "definition": "random(1..2#1)", "description": "<pre>power of x</pre>", "name": "n"}, "b2": {"group": "Original variables", "templateType": "randrange", "definition": "random(1..3#1)", "description": "<p>y range</p>", "name": "b2"}, "k3": {"group": "Combined variables", "templateType": "anything", "definition": "k1*k2", "description": "", "name": "k3"}, "a": {"group": "Original variables", "templateType": "anything", "definition": "random(0..(3-a2))", "description": "<p>start of x range {-5-c,5-c}</p>", "name": "a"}, "d": {"group": "Original variables", "templateType": "anything", "definition": "b+b2", "description": "<p>end&nbsp;of y range</p>", "name": "d"}, "c": {"group": "Original variables", "templateType": "anything", "definition": "a+a2", "description": "<p>end&nbsp;of x range</p>", "name": "c"}, "k": {"group": "Original variables", "templateType": "randrange", "definition": "random(2..5#1)", "description": "<p>constant in front of function</p>", "name": "k"}, "k5": {"group": "Combined variables", "templateType": "anything", "definition": "(d)^(m+1)-b^(m+1)", "description": "", "name": "k5"}}, "rulesets": {}, "parts": [{"correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": true, "mustBeReduced": true, "marks": "1", "minValue": "{k4*k5}", "allowFractions": true, "type": "numberentry", "showFeedbackIcon": true, "maxValue": "{k4*k5}", "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReducedPC": "50", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true}], "extensions": [], "statement": "<p>Evaluate the integral</p>\n<p>\\[&nbsp; \\int_{\\var{b}}^{\\var{d}}&nbsp; \\int_{\\var{a}}^{\\var{c}}&nbsp; \\var{k}&nbsp; \\simplify{x^{n}}&nbsp; \\simplify{y^{m}} \\, \\mathrm{d}x \\, \\mathrm{d}y \\,.&nbsp; \\]</p>", "advice": "<p>We can evaluate this iterated integral by straight-forward integration.</p>\n<p>First integrate with respect to $x$ while treating $\\simplify{y^{m}}$ as a constant.<br/>\\begin{align}<br/>\\int_{\\var{b}}^{\\var{d}}\\int_{\\var{a}}^{\\var{c}}\\var{k}\\simplify{x^{n}}\\simplify{y^{m}} &nbsp;\\,\\mathrm{d}x\\,\\mathrm{d}y <br/>&amp;=\\int_{\\var{b}}^{\\var{d}} \\, \\var{k}\\frac{1}{\\var{n+1}}x^{\\var{n+1}}\\bigg|_{x=\\var{a}}^{x=\\var{c}} \\, \\simplify{y^{m}}&nbsp; \\,\\mathrm{d}y \\\\<br/>&amp;=\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k1}}(\\var{c}^{\\var{n+1}}-\\var{a}^{\\var{n+1}})\\simplify{y^{m}}&nbsp; \\,\\mathrm{d}y \\\\<br/>&amp;=\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k1}}\\cdot \\simplify{{k2}y^{m}}&nbsp; \\,\\mathrm{d}y<br/>\\end{align}<br/>Now we can just integrate with respect to $y$ as usual.<br/>\\begin{align}<br/>\\int_{\\var{b}}^{\\var{d}}\\simplify[fractionNumbers]{{k3}}\\simplify{y^{m}}&nbsp; \\,\\mathrm{d}y<br/>&amp;= \\simplify[fractionNumbers]{{k3}}\\frac{1}{\\var{m+1}}y^{\\var{m+1}}\\bigg|_{y=\\var{b}}^{y=\\var{d}} <br/>= \\simplify[fractionNumbers]{{k4}}(\\var{d}^{\\var{m+1}}-\\var{b}^{\\var{m+1}}) <br/>= \\simplify[fractionNumbers]{{k4}}\\times\\var{k5} <br/>= \\simplify[fractionNumbers]{{k4*k5}}<br/>\\end{align}</p>", "variablesTest": {"condition": "2", "maxRuns": 100}, "tags": [], "ungrouped_variables": [], "type": "question", "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}]}], "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}