// Numbas version: exam_results_page_options
{"name": "Resolve a double integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"ans": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(upper-lower,3)", "description": "", "name": "ans"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "m"}, "fun": {"templateType": "anything", "group": "Ungrouped variables", "definition": "latex(\n  switch(\n    t=1,\n    '\\\\sin',\n    t=2,\n    '\\\\cos',\n    '\\\\exp'\n  )\n  +'(x^{'+m+'}+'+a+')'\n)", "description": "", "name": "fun"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "t"}, "upper": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-cos(1+a),t=2,sin(1+a),exp(1+a))", "description": "", "name": "upper"}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-cos(a),t=2,sin(a),exp(a))", "description": "", "name": "lower"}}, "ungrouped_variables": ["a", "upper", "lower", "m", "t", "ans", "fun"], "name": "Resolve a double integral", "functions": {}, "parts": [{"prompt": "<p>$I=\\;$[[0]]</p>\n<p>Input your answer to 3 decimal places.</p>", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "showCorrectAnswer": true, "allowFractions": false, "mustBeReduced": false, "correctAnswerFraction": false, "minValue": "ans", "maxValue": "ans", "precision": "3", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "strictPrecision": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "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>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "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>", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}