// Numbas version: exam_results_page_options
{"name": "Polar coordinates", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"type": "gapfill", "scripts": {}, "marks": 0, "prompt": "<p>Enter your answer correct to 3 decimal places.</p>\n<p>Answer=[[0]]</p>", "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "gaps": [{"strictPrecision": false, "precision": "3", "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "showPrecisionHint": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "minValue": "{a}*{pi}*(1-cos({t}*{pi}/{k}))/(2*{t})", "precisionPartialCredit": 0, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "customMarkingAlgorithm": "", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "maxValue": "{a}*{pi}*(1-cos({t}*{pi}/{k}))/(2*{t})", "variableReplacements": [], "allowFractions": false, "mustBeReducedPC": 0, "type": "numberentry"}]}], "ungrouped_variables": ["a", "k", "t"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "advice": "<p>\\(\\int_R\\int\\var{a}sin(\\var{t}(x^2+y^2))dxdy\\)</p>\n<p>Limits</p>\n<p style=\"padding-left: 30px;\">\\(x^2+y^2\\le\\frac{\\pi}{\\var{k}}\\) and \\(y\\ge0\\)</p>\n<p style=\"padding-left: 30px;\">This is the upper half of the circle centred on \\((0,0)\\) with radius \\(\\sqrt{\\frac{\\pi}{\\var{k}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=rcos(\\theta)\\)&nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; \\(y=rsin(\\theta)\\)</p>\n<p style=\"padding-left: 30px;\">\\(0\\le r\\le\\sqrt{\\frac{\\pi}{\\var{k}}}\\)&nbsp; &nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp; &nbsp; \\(0\\le \\theta\\le\\pi\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\">\\(dxdy=rdrd\\theta\\)</p>\n<p>The function</p>\n<p style=\"padding-left: 30px;\">\\(x^2+y^2=r^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\implies&nbsp;\\var{a}sin(\\var{t}(x^2+y^2))=\\var{a}sin(\\var{t}r^2)\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\">\\(\\int_R\\int\\var{a}sin(\\var{t}(x^2+y^2))dxdy=\\int_0^{\\pi}\\int_0^{\\sqrt{\\frac{\\pi}{\\var{k}}}}\\var{a}sin(\\var{t}r^2)rdrd\\theta\\)</p>\n<p>Inner integral</p>\n<p style=\"padding-left: 30px;\">\\(\\int_0^{\\sqrt{\\frac{\\pi}{\\var{k}}}}\\var{a}sin(\\var{t}r^2)rdr=-\\frac{\\var{a}}{\\simplify{{t}*2}}cos(\\var{t}r^2)\\big|_0^{\\sqrt{\\frac{\\pi}{\\var{k}}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{a}}{\\simplify{{t}*2}}cos(\\frac{\\var{t}\\pi}{\\var{k}})-\\left(-\\frac{\\var{a}}{\\simplify{{t}*2}}cos(0)\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=-\\frac{\\var{a}}{\\simplify{{t}*2}}cos(\\frac{\\var{t}\\pi}{\\var{k}})+\\frac{\\var{a}}{\\simplify{{t}*2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{a}}{\\simplify{{t}*2}}\\left(1-cos(\\frac{\\var{t}\\pi}{\\var{k}})\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}*(1-cos({t}*{pi}/{k}))/(2*{t})}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Outer Integral</p>\n<p style=\"padding-left: 30px;\">\\(\\int_0^{\\pi}\\simplify{{a}*(1-cos({t}*{pi}/{k}))/(2*{t})}d\\theta\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}*(1-cos({t}*{pi}/{k}))/(2*{t})}\\theta\\big|_0^\\var{pi}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}*(1-cos({t}*{pi}/{k}))/(2*{t})}{\\pi}-0\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}*{pi}*(1-cos({t}*{pi}/{k}))/(2*{t})}\\)</p>", "rulesets": {}, "tags": [], "preamble": {"js": "", "css": ""}, "variable_groups": [], "variables": {"t": {"templateType": "randrange", "group": "Ungrouped variables", "name": "t", "definition": "random(2..8#1)", "description": ""}, "a": {"templateType": "randrange", "group": "Ungrouped variables", "name": "a", "definition": "random(2..16#1)", "description": ""}, "k": {"templateType": "randrange", "group": "Ungrouped variables", "name": "k", "definition": "random(2..10#1)", "description": ""}}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "name": "Polar coordinates", "statement": "<p>Evaluate the integral below using polar co-ordinates:</p>\n<p style=\"padding-left: 30px;\">\\(\\int\\int_R\\var{a}sin(\\var{t}(x^2+y^2))dxdy\\)</p>\n<p style=\"padding-left: 30px;\">where \\(R\\) is the region of the plane enclosed by the circle \\(x^2+y^2\\le\\frac{\\pi}{\\var{k}}\\) and \\(y\\ge0\\).</p>\n<p style=\"padding-left: 30px;\"></p>", "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/"}]}