// Numbas version: exam_results_page_options
{"name": "Evaluate double integrals with numerical limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "b", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "c", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "d", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "name": "g", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "name": "Evaluate double integrals with numerical limits", "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"prompt": "<p>\\[I = \\int^\\var{a}_{y=1} \\int^\\var{b}_{x=0} \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\]</p>\n<p>$I =$&nbsp;[[0]]</p>", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers in your answer as integers or fractions, not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>\\[ I = \\int^\\pi_{x=0} \\int^\\var{h}_{y=0} \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\]</p>\n<p>$I=$&nbsp;[[0]]</p>", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{-h^(f+1)*((-1)^g-1)/(g*(f+1))}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "answersimplification": "fractionnumbers", "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>Evaluate the following double integrals.</p>\n    <p>Input your answer as an integer or a fraction, not as a decimal.</p>", "tags": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Two double integrals with numerical limits</p>"}, "type": "question", "extensions": [], "advice": "<h4>(a)</h4>\n<p>We proceed to evaluate the double-integral:</p>\n<p>\\begin{align}<br/>I &amp;= \\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\\\ <br/>&amp;= \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]_{x=0}^\\var{b} \\; \\mathrm{d}y \\\\ <br/>&amp;= \\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right) \\; \\mathrm{d}y \\\\ <br/>&amp;= \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1 \\; \\mathrm{d}y \\\\ <br/>&amp;= \\simplify[std]{{c*b*a}+{d*b^2*a^2}-{c*b}-{d*b^2}} \\\\ <br/>&amp;=&nbsp;\\simplify[std]{{(c*b*a)+(d*b^2*a^2)-(c*b)-(d*b^2)}}<br/>\\end{align}</p>\n<h4>(b)</h4>\n<p>\\begin{align}<br/>I &amp;= \\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\\\ <br/>&amp;= \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]_{y=0}^\\var{h} \\; \\mathrm{d}x \\\\[0.5em]<br/> &amp;= \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)} \\; \\mathrm{d}x&nbsp; \\\\[0.5em]<br/> &amp;= \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0&nbsp; \\\\[0.5em]<br/>&amp;= -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\[0.5em]<br/>&amp;= \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}}<br/>\\end{align}</p>", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}