// Numbas version: exam_results_page_options
{"name": "Hina's copy of 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": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "name": "Hina's copy of Evaluate double integrals with numerical limits, ", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>(a) We proceed to evaluate the double-integral:</p>\n<p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}xy} \\right)&nbsp; dx&nbsp; dy \\\\ &amp;=&amp; \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]^\\var{b}_0&nbsp; dy \\\\ &amp;=&amp;\\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right)&nbsp; dy \\\\ &amp;=&amp; \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1&nbsp; dy \\\\ &amp;=&amp;\\simplify[std]{{c*b*a}+{d*b^2*a^2}-{c*b}-{d*b^2}} \\\\ &amp;=&amp;\\simplify[std]{{(c*b*a)+(d*b^2*a^2)-(c*b)-(d*b^2)}}\\end{eqnarray*}\\]</p>\n<p>(b) \\[\\begin{eqnarray*} I&amp;=&amp;\\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} dy&nbsp; dx \\\\ &amp;=&amp; \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]^\\var{h}_0 dx \\\\ &amp;=&amp; \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)}&nbsp; dx&nbsp; \\\\ &amp;=&amp; \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0&nbsp; \\\\ &amp;=&amp; -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\ &amp;=&amp; \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}}\\end{eqnarray*}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>\\[I=\\int^\\var{a}_{y=1} \\int^\\var{b}_{x=0} \\left(\\var{c}+\\simplify[std]{{4*d}xy} \\right) &nbsp;dx\\, &nbsp;dy \\]</p>\n<p>$I=\\;\\;$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "alwaysreplace", "gaps": [{"notallowed": {"message": "<p>Input all numbers in your answer as integers or fractions, not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "marks": 4, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>\\[I=\\int^\\pi_{x=0} \\int^\\var{h}_{y=0} \\simplify[std]{y^{f}sin({g}x)} dy \\, dx \\]</p>\n<p>$I=\\;\\;$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "fractionnumbers", "scripts": {}, "answer": "{-h^(f+1)*((-1)^g-1)/(g*(f+1))}", "marks": 4, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Evaluate the following double integrals.</p>\n<p>Input your answer as an integer or a fraction, not as a decimal.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(1..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}}, "metadata": {"description": "<p>Double integrals (2) with numerical limits</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Hina Ahmed", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1160/"}]}]}], "contributors": [{"name": "Hina Ahmed", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1160/"}]}