// 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": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "d", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "b", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "name": "g", "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": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "Evaluate double integrals with numerical limits, ", "functions": {}, "variable_groups": [], "parts": [{"scripts": {}, "gaps": [{"answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Input all numbers in your answer as integers or fractions, not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "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>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-h^(f+1)*((-1)^g-1)/(g*(f+1))}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "fractionnumbers", "type": "jme", "showCorrectAnswer": true, "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "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>", "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": ["Calculus", "calculus", "checked2015", "double integral", "MAS1603", "MAS2104", "tested1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>30/06/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Minor change to prompt.</p>\n    <p><strong>19/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p>Did not add Show steps.</p>\n    <p>Checked calculation.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p><strong>22/12/2012:(WHF)</strong></p>\n    <p>Corrected mistake in last part, the upper limit in the integral was set as the value of a which was the upper limit in the first part, but it should have been the value of h.</p>\n    <p>Checked calculations, OK. Added tested1 tag.</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>\n    <p>Question appears to be working correctly.</p>\n    <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Double integrals (2) with numerical limits</p>"}, "showQuestionGroupNames": false, "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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}