// Numbas version: exam_results_page_options
{"name": "Simple integrals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "rulesets": {}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "prompt": "<p>Insert your answer correct to three decimal places</p>\n<p>Answer =&nbsp;[[0]]</p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "gaps": [{"showCorrectAnswer": true, "scripts": {}, "strictPrecision": false, "precisionType": "dp", "maxValue": "{k3}*({a2}^({m}+1)-{a1}^({m}+1))*({b2}^({n}+1)-{b1}^({n}+1))/(({n}+1)*({m}+1))+{k1}*({b2}-{b1})*({a2}^2-{a1}^2)/2-{k2}*({a2}-{a1})*({b2}^2-{b1}^2)/2", "showFeedbackIcon": true, "variableReplacements": [], "marks": 1, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{k3}*({a2}^({m}+1)-{a1}^({m}+1))*({b2}^({n}+1)-{b1}^({n}+1))/(({n}+1)*({m}+1))+{k1}*({b2}-{b1})*({a2}^2-{a1}^2)/2-{k2}*({a2}-{a1})*({b2}^2-{b1}^2)/2", "mustBeReduced": false, "mustBeReducedPC": 0, "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": "75", "type": "numberentry", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}], "advice": "<p>\\(\\int_\\var{a1}^\\var{a2}\\int_\\var{b1}^\\var{b2}\\left(\\var{k3}x^\\var{m}y^\\var{n}+\\var{k1}x-\\var{k2}y\\right)dy\\,dx\\)</p>\n<p>Inner integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{b1}^\\var{b2}\\left(\\var{k3}x^\\var{m}y^\\var{n}+\\var{k1}x-\\var{k2}y\\right)dy\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{k3}x^\\var{m}y^\\simplify{{n}+1}}{\\simplify{{n}+1}}+\\var{k1}xy-\\frac{\\var{k2}y^2}{2}\\big|_\\var{b1}^\\var{b2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{k3}x^\\var{m}(\\var{b2})^\\simplify{{n}+1}}{\\simplify{{n}+1}}+\\var{k1}x(\\var{b2})-\\frac{\\var{k2}\\left(\\var{b2}^2\\right)}{2}-\\frac{\\var{k3}x^\\var{m}(\\var{b1})^\\simplify{{n}+1}}{\\simplify{{n}+1}}-\\var{k1}x(\\var{b1})+\\frac{\\var{k2}\\left(\\var{b1}^2\\right)}{2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))}}{\\simplify{{n}+1}}x^\\var{m}+\\simplify{{k1}*({b2}-{b1})}x-\\frac{\\simplify{{k2}*({b2}^2-{b1}^2)}}{2}\\)</p>\n<p>Outer integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a1}^\\var{a2}\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))}}{\\simplify{{n}+1}}x^\\var{m}+\\simplify{{k1}*({b2}-{b1})}x-\\frac{\\simplify{{k2}*({b2}^2-{b1}^2)}}{2}dx\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))}}{\\simplify{{n}+1}}\\frac{x^\\simplify{{m}+1}}{\\simplify{{m}+1}}+\\simplify{{k1}*({b2}-{b1})}\\frac{x^2}{2}-\\frac{\\simplify{{k2}*({b2}^2-{b1}^2)}}{2}x\\big|_\\var{a1}^\\var{a2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))}}{\\simplify{{n}+1}}\\frac{(\\var{a2})^\\simplify{{m}+1}}{\\simplify{{m}+1}}+\\simplify{{k1}*({b2}-{b1})}\\frac{(\\var{a2})^2}{2}-\\frac{\\simplify{{k2}*({b2}^2-{b1}^2)}}{2}(\\var{a2})-\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))}}{\\simplify{{n}+1}}\\frac{(\\var{a1})^\\simplify{{m}+1}}{\\simplify{{m}+1}}-\\simplify{{k1}*({b2}-{b1})}\\frac{(\\var{a1})^2}{2}+\\frac{\\simplify{{k2}*({b2}^2-{b1}^2)}}{2}(\\var{a1})\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{k3}*({b2}^({n}+1)-{b1}^({n}+1))*({a2}^{{m}+1}-{a1}^{{m}+1})}}{\\simplify{({n}+1)*({m}+1)}}+\\simplify{{k1}*({b2}-{b1})*({a2}^2-{a1}^2)/2}-\\simplify{{k2}*({b2}^2-{b1}^2)*({a2}-{a1})/2}\\)</p>", "ungrouped_variables": ["a2", "a1", "b1", "b2", "m", "n", "k1", "k2", "k3"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Evaluate the integral:</p>\n<p style=\"padding-left: 90px;\">\\(\\int_\\var{a1}^\\var{a2}\\int_\\var{b1}^\\var{b2}\\left(\\var{k3}x^\\var{m}y^\\var{n}+\\var{k1}x-\\var{k2}y\\right)dy\\,dx\\)</p>", "extensions": [], "variable_groups": [], "variables": {"n": {"name": "n", "description": "", "group": "Ungrouped variables", "definition": "random(2..5#1)", "templateType": "randrange"}, "b2": {"name": "b2", "description": "", "group": "Ungrouped variables", "definition": "random(3..5#1)", "templateType": "randrange"}, "a2": {"name": "a2", "description": "", "group": "Ungrouped variables", "definition": "random(3..5#1)", "templateType": "randrange"}, "m": {"name": "m", "description": "", "group": "Ungrouped variables", "definition": "random(2..4#1)", "templateType": "randrange"}, "k2": {"name": "k2", "description": "", "group": "Ungrouped variables", "definition": "random(1..10#1)", "templateType": "randrange"}, "k1": {"name": "k1", "description": "", "group": "Ungrouped variables", "definition": "random(1..10#1)", "templateType": "randrange"}, "k3": {"name": "k3", "description": "", "group": "Ungrouped variables", "definition": "random(2..5#1)", "templateType": "randrange"}, "b1": {"name": "b1", "description": "", "group": "Ungrouped variables", "definition": "random(0..2#1)", "templateType": "randrange"}, "a1": {"name": "a1", "description": "", "group": "Ungrouped variables", "definition": "random(0..2#1)", "templateType": "randrange"}}, "name": "Simple integrals", "tags": [], "preamble": {"js": "", "css": ""}, "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/"}]}