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{"name": "Reverse the order of integration#2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variable_groups": [], "rulesets": {}, "preamble": {"js": "", "css": ""}, "advice": "<p>\\(\\int_0^\\var{a}\\int_{\\var{k}y}^{\\simplify{{k}*{a}}}\\var{b}sin(\\var{c}x^2)dx\\,dy\\)</p>\n<p>The limits are four lines \\(x=\\var{k}y,\\,\\,x=\\simplify{{k}*{a}}\\) ,&nbsp; \\(y=0\\) and&nbsp; \\( y=\\var{a}\\)</p>\n<p>These limits form a triangle, so we must keep:&nbsp;\\(x=\\simplify{{k}*{a}},\\,\\,x=\\var{k}y\\)&nbsp; &nbsp;and&nbsp; &nbsp;\\(y=0\\)</p>\n<p>On the inner integral we will have \\(y=0\\) and \\(y=\\frac{x}{\\var{k}}\\)</p>\n<p>On the outer integral we have&nbsp;\\(x=\\simplify{{k}*{a}}\\) and to this we add \\(x=0\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p></p>\n<p>\\(\\int_0^\\simplify{{a}*{k}}\\int_0^{\\frac{x}{\\var{k}}}\\var{b}sin(\\var{c}x^2)dy\\,dx\\)</p>\n<p></p>\n<p></p>\n<p>Inner integral:&nbsp; &nbsp; \\(\\int_0^{\\frac{x}{\\var{k}}}\\var{b}sin(\\var{c}x^2)dy\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\var{b}sin(\\var{c}x^2).y\\,\\big|_0^{\\frac{x}{\\var{k}}}\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\var{b}sin(\\var{c}x^2)*\\frac{x}{\\var{k}}-\\var{b}sin(\\var{c}x^2)*0\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\frac{\\var{b}}{\\var{k}}xsin(\\var{c}x^2)\\)</p>\n<p>Outer integral:&nbsp; &nbsp; \\(\\int_0^\\simplify{{a}*{k}}\\frac{\\var{b}}{\\var{k}}xsin(\\var{c}x^2)dx\\)</p>\n<p style=\"padding-left: 120px;\">This requires a u-substitution. Let \\(u=\\var{c}x^2\\)&nbsp; &nbsp; &nbsp; &nbsp;\\(\\frac{du}{dx}=\\simplify{2*{c}}x\\)</p>\n<p style=\"padding-left: 480px;\">\\(du=\\simplify{2*{c}}x\\,dx\\)</p>\n<p style=\"padding-left: 480px;\">\\(\\frac{1}{\\simplify{2*{c}}x}du=dx\\)</p>\n<p style=\"padding-left: 120px;\">\\(\\implies\\int\\frac{\\var{b}}{\\var{k}}xsin(u)\\frac{1}{\\simplify{2*{c}}x}du\\)</p>\n<p style=\"padding-left: 120px;\"></p>\n<p style=\"padding-left: 120px;\">\\(=\\int\\frac{\\var{b}}{\\simplify{{k}*2*{c}}}sin(u)du\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\simplify{{b}/({k}*2*{c})}cos(u)\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\simplify{{b}/({k}*2*{c})}cos(\\var{c}x^2)\\,\\big|_0^\\simplify{{a}*{k}}\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\simplify{{b}/({k}*2*{c})}cos(\\simplify{{c}*{a}^2*{k}^2})+\\simplify{{b}/({k}*2*{c})}cos(0)\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\simplify{{b}/({k}*2*{c})}cos(\\simplify{{c}*{a}^2*{k}^2})+\\simplify{{b}/({k}*2*{c})}\\)</p>\n<p style=\"padding-left: 120px;\"></p>\n<p style=\"padding-left: 120px;\"></p>", "parts": [{"scripts": {}, "type": "gapfill", "prompt": "<p>Enter your answer correct to 3 decimal places.</p>\n<p>Answer =&nbsp;[[0]]</p>", "variableReplacements": [], "gaps": [{"type": "numberentry", "precision": "3", "precisionPartialCredit": 0, "minValue": "{b}*(1-cos(angle))/(2*{c}*{k})", "allowFractions": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "mustBeReducedPC": 0, "precisionType": "dp", "strictPrecision": false, "scripts": {}, "marks": 1, "variableReplacements": [], "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": false, "maxValue": "{b}*(1-cos(angle))/(2*{c}*{k})", "correctAnswerStyle": "plain"}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true}], "name": "Reverse the order of integration#2", "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["a", "b", "c", "k", "angle"], "tags": [], "functions": {}, "statement": "<p>Evaluate the integral below by reversing the order of integration.</p>\n<p style=\"padding-left: 90px;\">\\(\\int_0^\\var{a}\\int_{\\var{k}y}^{\\simplify{{k}*{a}}}\\var{b}sin(\\var{c}x^2)dx\\,dy\\)</p>", "variables": {"b": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(4..16#1)", "name": "b"}, "k": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(2..8#1)", "name": "k"}, "a": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(1..5#1)", "name": "a"}, "angle": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "{c}*{k}^2*{a}^2", "name": "angle"}, "c": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(2..12#1)", "name": "c"}}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}