// Numbas version: exam_results_page_options
{"name": "Simple integrals #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": false, "showFeedbackIcon": true, "maxValue": "{frac}*{factor1}*{factor2}", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "scripts": {}, "minValue": "{frac}*{factor1}*{factor2}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "precisionPartialCredit": "75", "marks": 1, "precision": "2", "precisionType": "dp", "showPrecisionHint": false, "type": "numberentry", "strictPrecision": false, "mustBeReduced": false}], "type": "gapfill", "variableReplacements": [], "prompt": "<p>Insert your answer correct to two decimal places</p>\n<p>Answer =&nbsp;[[0]]</p>"}], "statement": "<p>Evaluate the integral:</p>\n<p style=\"padding-left: 90px;\">\\(\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k1}e^{\\var{k2}x+\\var{k3}y}dx\\,dy\\)</p>", "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "ungrouped_variables": ["k1", "k2", "k3", "a1", "b1", "a2", "b2", "frac", "factor1", "factor2", "k4", "s1", "s2", "s3", "s4"], "advice": "<p>\\(\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k1}e^{\\var{k2}x+\\var{k3}y}dx\\,dy\\)</p>\n<p>Inner integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a1}^\\var{b1}\\var{k1}e^{\\var{k2}x+\\var{k3}y}dx\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{k1}}{\\var{k2}}e^{\\var{k2}x+\\var{k3}y}\\big|_\\var{a1}^\\var{b1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\var{k4}}e^{\\var{k2}x+\\var{k3}y}\\big|_\\var{a1}^\\var{b1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{b1}}+\\var{k3}y}-\\left(-\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{a1}}+\\var{k3}y}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{b1}}+\\var{k3}y}+\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{a1}}+\\var{k3}y}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Outer integral:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a2}^\\var{b2}-\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{b1}}+\\var{k3}y}+\\frac{\\var{k1}}{\\var{k4}}e^{\\simplify{{k2}*{a1}}+\\var{k3}y}dy\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\var{k4}}\\frac{1}{\\var{k3}}e^{\\simplify{{k2}*{b1}}+\\var{k3}y}+\\frac{\\var{k1}}{\\var{k4}}\\frac{1}{\\var{k3}}e^{\\simplify{{k2}*{a1}}+\\var{k3}y}\\big|_\\var{a2}^\\var{b2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{b1}}+\\simplify{{k3}*{b2}}}+\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{a1}}+\\simplify{{k3}*{b2}}}-\\left(-\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{b1}}+\\simplify{{k3}*{a2}}}+\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{a1}}+\\simplify{{k3}*{a2}}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{b1}+{k3}*{b2}}}+\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{a1}+{k3}*{b2}}}+\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{b1}+{k3}*{a2}}}-\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}e^{\\simplify{{k2}*{a1}+{k3}*{a2}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{k1}}{\\simplify{{k4}*{k3}}}\\left(e^{\\simplify{{k2}*{b1}+{k3}*{b2}}}-e^{\\simplify{{k2}*{a1}+{k3}*{b2}}}-e^{\\simplify{{k2}*{b1}+{k3}*{a2}}}+e^{\\simplify{{k2}*{a1}+{k3}*{a2}}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{k1}/({k2}*{k3})}\\left(\\var{s1}-\\var{s2}-\\var{s3}+\\var{s4}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{frac}*{factor1}*{factor2}}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>", "tags": [], "rulesets": {}, "name": "Simple integrals #2", "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "variables": {"factor2": {"name": "factor2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k3}*{b2})-e^({k3}*{a2})"}, "b1": {"name": "b1", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(1.2..1.9#0.1)"}, "s2": {"name": "s2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k2}*{a1}+{k3}*{b2})"}, "k2": {"name": "k2", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(-8..-2#1)"}, "k3": {"name": "k3", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..5#1)"}, "k4": {"name": "k4", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "-k2"}, "k1": {"name": "k1", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..10#1)"}, "factor1": {"name": "factor1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k2}*{b1})-e^({k2}*{a1})"}, "frac": {"name": "frac", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "{k1}/({k2}*{k3})"}, "s3": {"name": "s3", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k2}*{b1}+{k3}*{a2})"}, "b2": {"name": "b2", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(1..2#0.2)"}, "s4": {"name": "s4", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k2}*{a1}+{k3}*{a2})"}, "a1": {"name": "a1", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(0.4..0.9#0.1)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "e^({k2}*{b1}+{k3}*{b2})"}, "a2": {"name": "a2", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(0..0.9#0.1)"}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "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/"}]}