// Numbas version: exam_results_page_options
{"name": "Luis's copy of Resolve a double integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "precision": "3", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "precisionPartialCredit": 0, "scripts": {}, "minValue": "ans", "type": "numberentry", "maxValue": "ans", "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "marks": 1, "strictPrecision": true, "precisionType": "dp", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision."}], "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$I=\\;$[[0]]</p>\n<p>Input your answer to 3 decimal places.</p>", "scripts": {}}], "name": "Luis's copy of Resolve a double integral", "ungrouped_variables": ["a", "upper", "lower", "m", "t", "ans", "fun"], "variables": {"a": {"description": "", "definition": "random(1..9)", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "t": {"description": "", "definition": "random(1..3)", "templateType": "anything", "name": "t", "group": "Ungrouped variables"}, "ans": {"description": "", "definition": "precround(upper-lower,3)", "templateType": "anything", "name": "ans", "group": "Ungrouped variables"}, "fun": {"description": "", "definition": "latex(\n  switch(\n    t=1,\n    '\\\\sin',\n    t=2,\n    '\\\\cos',\n    '\\\\exp'\n  )\n  +'(x^{'+m+'}+'+a+')'\n)", "templateType": "anything", "name": "fun", "group": "Ungrouped variables"}, "upper": {"description": "", "definition": "switch(t=1,-cos(1+a),t=2,sin(1+a),exp(1+a))", "templateType": "anything", "name": "upper", "group": "Ungrouped variables"}, "lower": {"description": "", "definition": "switch(t=1,-cos(a),t=2,sin(a),exp(a))", "templateType": "anything", "name": "lower", "group": "Ungrouped variables"}, "m": {"description": "", "definition": "random(2..5)", "templateType": "anything", "name": "m", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "statement": "<p>Evaluate the following repeated integral:</p>\n<p>\\[ I = \\int_0^1 \\; \\mathrm{d}x \\; \\int_0^{\\simplify[all]{x^{m-1}}}\\var{m} \\var{fun} \\; \\mathrm{d}y \\]</p>", "metadata": {"description": "<p>Calculate a repeated integral of the form $\\displaystyle I=\\int_0^1\\;dx\\;\\int_0^{x^{m-1}}mf(x^m+a)dy$</p>\n<p>The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "functions": {}, "variable_groups": [], "advice": "<p>We want to find</p>\n<p>\\[ I=\\int_0^1 \\; dx \\; \\int_0^{\\simplify[all]{x^{m-1}}} \\var{m} \\var{fun} \\; \\mathrm{d}y \\]</p>\n<p>The innermost integral gives:</p>\n<p>\\[ \\int_0^{\\simplify[all]{x^{m-1}}}\\var{m} \\var{fun} \\; \\mathrm{d}y = \\left[\\var{m}y \\; \\var{fun} \\right]_0^{\\simplify[all]{x^{m-1}}}=\\simplify[all]{{m}x^{m-1}} \\var{fun} \\]</p>\n<p>So we have to find &nbsp;$\\displaystyle I=\\int_0^1\\simplify[all]{{m}x^{m-1}} \\var{fun} \\; \\mathrm{d}x$.</p>\n<p>Note that if we use the substitution $u=\\simplify[all]{x^{m}+{a}}$ then it is easy to find this last definite integral and we find that:</p>\n<p>$I=\\var{ans}$ to 3 decimal places.</p>", "extensions": [], "tags": [], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}