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{"name": "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": "switch(t=1,'$\\\\sin(x^{\\\\var{m}}+\\\\var{a})$',t=2,'$\\\\cos(x^{\\\\var{m}}+\\\\var{a})$','$\\\\exp(x^{\\\\var{m}}+\\\\var{a})$')", "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"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Double integral, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans+0.001", "minValue": "ans-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\displaystyle I=\\int_0^1\\;dx\\;\\int_0^{\\simplify[all]{x^{m-1}}}\\var{m}${fun}$dy$</p>\n      <p>$I=\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Evaluate the following repeated integral:</p>", "tags": ["checked2015", "MAS1603", "MAS2304"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>01/02/2013:</strong></p>\n            <p>First draft completed.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Repeated integral of the form: $\\displaystyle I=\\int_0^1\\;dx\\;\\int_0^{x^{m-1}}mf(x^m+a)dy$</p>\n<p>&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>We want to find</p>\n    <p>&nbsp;</p>\n    <p>$\\displaystyle I=\\int_0^1\\;dx\\;\\int_0^{\\simplify[all]{x^{m-1}}}\\var{m}${fun}$dy$</p>\n    <p>The innermost integral gives:</p>\n    <p>$\\displaystyle \\int_0^{\\simplify[all]{x^{m-1}}}\\var{m}${fun}$dy=\\left[\\var{m}y\\;\\right.${fun}$\\displaystyle \\left. \\right]_0^{\\simplify[all]{x^{m-1}}}=\\simplify[all]{{m}x^{m-1}}${fun}</p>\n    <p>So we have to find &nbsp;$\\displaystyle I=\\int_0^1\\simplify[all]{{m}x^{m-1}}${fun}$dx$</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>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}