// Numbas version: finer_feedback_settings
{"name": "Reverse the order of integration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a1", "a2", "b1", "c1", "d1", "k1", "k2", "k3"], "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>\\(\\int_\\var{b1}^\\var{a2}\\int_{x^2}^{\\var{c1}x+\\var{d1}}\\left(\\var{k1}xy+\\var{k2}x+\\var{k3}y\\right)dy\\,dx\\)</p>\n<p>The limits are \\(x=\\var{b1},\\,\\,x=\\var{a2}\\) the curve&nbsp; \\(y=x^2\\) and the straight line \\( y=\\var{c1}x+\\var{d1}\\)</p>\n<p>We must find their points of intersection: \\(x^2=\\var{c1}x+\\var{d1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\implies x^2-\\var{c1}x-\\var{d1}=0\\)</p>\n<p style=\"padding-left: 30px;\">Solving this quadratic gives&nbsp;\\(x=\\var{b1},\\,\\,x=\\var{a2}\\)</p>\n<p style=\"padding-left: 30px;\">This allows us to sketch our area of integration.</p>\n<p>We must keep \\(y=x^2\\,\\,and\\,\\,y=\\var{c1}x+\\var{d1}\\) but rearrange them to \\(x=\\sqrt{y}\\) and \\(x=\\frac{y-\\var{d1}}{\\var{c1}}\\)</p>\n<p></p>\n<p>We also need two horizontal lines that will touch the area without bisecting it. These are provided by \\(y=0\\) and \\(y=\\simplify{{a2}^2}\\)</p>\n<p>\\(\\int_0^\\simplify{{a2}^2}\\int_\\frac{y-\\var{d1}}{\\var{c1}}^{\\sqrt{y}}\\left(\\var{k1}xy+\\var{k2}x+\\var{k3}y\\right)dx\\,dy\\)</p>\n<p>Inner integral:&nbsp; &nbsp; &nbsp;\\(\\int_\\frac{y-\\var{d1}}{\\var{c1}}^{\\sqrt{y}}\\left(\\var{k1}xy+\\var{k2}x+\\var{k3}y\\right)dx\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\frac{\\var{k1}x^2y}{2}+\\frac{\\var{k2}x^2}{2}+\\var{k3}yx\\big|_\\frac{y-\\var{d1}}{\\var{c1}}^{\\sqrt{y}}\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\frac{\\var{k1}(\\sqrt{y})^2y}{2}+\\frac{\\var{k2}(\\sqrt{y})^2}{2}+\\var{k3}y(\\sqrt{y})-\\frac{\\var{k1}}{2}{(\\frac{y-\\var{d1}}{\\var{c1}})^2y}-\\frac{\\var{k2}}{2}(\\frac{y-\\var{d1}}{\\var{c1}})^2-\\var{k3}y(\\frac{y-\\var{d1}}{\\var{c1}})\\)</p>\n<p style=\"padding-left: 120px;\">\\(=\\frac{\\var{k1}}{2}y^2+\\frac{\\var{k2}}{2}y+\\var{k3}y^{1.5}-\\frac{\\var{k1}}{\\simplify{2*{c1}^2}}(y^2-\\simplify{2*{d1}}y+\\simplify{{d1}^2})y-\\frac{\\var{k2}}{\\simplify{2*{c1}^2}}(y^2-\\simplify{2*{d1}}y+\\simplify{{d1}^2})-\\frac{\\var{k3}}{\\simplify{{c1}}}(y^2-\\var{d1}y)\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\frac{\\var{k1}}{\\simplify{2*{c1}^2}}y^3+\\frac{\\simplify{{k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1}}}{\\simplify{2*{c1}^2}}y^2+\\frac{\\simplify{{k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1}}}{\\simplify{2*{c1}^2}}y+\\var{k3}y^{1.5}-\\simplify{{k2}*{d1}^2/(2*{c1}^2)}\\)</p>\n<p>Outer integral:&nbsp; &nbsp;&nbsp;</p>\n<p style=\"padding-left: 120px;\">\\(\\int_0^{\\simplify{{a2}^2}}-\\frac{\\var{k1}}{\\simplify{2*{c1}^2}}y^3+\\frac{\\simplify{{k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1}}}{\\simplify{2*{c1}^2}}y^2+\\frac{\\simplify{{k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1}}}{\\simplify{2*{c1}^2}}y+\\var{k3}y^{1.5}-\\simplify{{k2}*{d1}^2/(2*{c1}^2)}dy\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\frac{\\var{k1}}{\\simplify{8*{c1}^2}}y^4+\\frac{\\simplify{{k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1}}}{\\simplify{6*{c1}^2}}y^3+\\frac{\\simplify{{k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1}}}{\\simplify{4*{c1}^2}}y^2+\\frac{\\var{k3}}{2.5}y^{2.5}-\\simplify{{k2}*{d1}^2/(2*{c1}^2)}y\\,\\,\\big|_0^{\\simplify{{a2}^2}}\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\frac{\\var{k1}}{\\simplify{8*{c1}^2}}(\\simplify{{a2}^2})^4+\\frac{\\simplify{{k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1}}}{\\simplify{6*{c1}^2}}(\\simplify{{a2}^2})^3+\\frac{\\simplify{{k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1}}}{\\simplify{4*{c1}^2}}(\\simplify{{a2}^2})^2+\\frac{\\var{k3}}{2.5}(\\simplify{{a2}^2})^{2.5}-\\simplify{{k2}*{d1}^2/(2*{c1}^2)}(\\simplify{{a2}^2})-0\\)</p>\n<p style=\"padding-left: 120px;\">\\(=-\\frac{\\simplify{{k1}*{a2}^8}}{\\simplify{8*{c1}^2}}+\\frac{\\simplify{({k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1})*{a2}^6}}{\\simplify{6*{c1}^2}}+\\frac{\\simplify{({k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1})*{a2}^4}}{\\simplify{4*{c1}^2}}+\\simplify{{k3}*{a2}^5/2.5}-\\simplify{{k2}*{d1}^2*{a2}^2/(2*{c1}^2)}-0\\)</p>\n<p style=\"padding-left: 120px;\"></p>\n<p style=\"padding-left: 120px;\"></p>", "variable_groups": [], "statement": "<p>Evaluate the integral below by reversing the order of integration.</p>\n<p style=\"padding-left: 90px;\">\\(\\int_\\var{b1}^\\var{a2}\\int_{x^2}^{\\var{c1}x+\\var{d1}}\\left(\\var{k1}xy+\\var{k2}x+\\var{k3}y\\right)dy\\,dx\\)</p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "functions": {}, "name": "Reverse the order of integration", "tags": [], "variables": {"a1": {"name": "a1", "definition": "random(1..3#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "d1": {"name": "d1", "definition": "a1*a2", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a2": {"name": "a2", "definition": "random(4..6#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k1": {"name": "k1", "definition": "random(2..7#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k3": {"name": "k3", "definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b1": {"name": "b1", "definition": "-a1", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "k2": {"name": "k2", "definition": "random(2..8#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "c1": {"name": "c1", "definition": "a2-a1", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Enter your answer correct to 3 decimal places.</p>\n<p>Answer =&nbsp;[[0]]</p>", "scripts": {}, "type": "gapfill", "gaps": [{"mustBeReducedPC": 0, "precisionType": "dp", "correctAnswerFraction": false, "allowFractions": false, "showCorrectAnswer": true, "maxValue": "-{k1}*{a2}^8/(8*{c1}^2)+({k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1})*{a2}^6/(6*{c1}^2)+({k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1})*{a2}^4/(4*{c1}^2)+{k3}*{a2}^5/2.5-{k2}*{d1}^2*{a2}^2/(2*{c1}^2)", "showPrecisionHint": false, "type": "numberentry", "mustBeReduced": false, "marks": 1, "strictPrecision": false, "variableReplacements": [], "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "3", "minValue": "-{k1}*{a2}^8/(8*{c1}^2)+({k1}*{c1}^2+2*{d1}*{k1}-{k2}-2*{k3}*{c1})*{a2}^6/(6*{c1}^2)+({k2}*{c1}^2-{k1}*{d1}^2+2*{k2}*{d1}+2*{k3}*{d1}*{c1})*{a2}^4/(4*{c1}^2)+{k3}*{a2}^5/2.5-{k2}*{d1}^2*{a2}^2/(2*{c1}^2)", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision."}], "variableReplacementStrategy": "originalfirst"}], "preamble": {"js": "", "css": ""}, "extensions": [], "rulesets": {}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "resources": []}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}