// Numbas version: exam_results_page_options
{"name": "Polar coordinates #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "tags": [], "name": "Polar coordinates #2", "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "ungrouped_variables": ["a", "k", "t", "k2", "c"], "parts": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "unitTests": [], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "variableReplacements": [], "minValue": "{a}*{c}*2*{pi}/{t}", "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "precisionPartialCredit": 0, "precisionType": "dp", "type": "numberentry", "correctAnswerFraction": false, "precision": "3", "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "customMarkingAlgorithm": "", "strictPrecision": false, "maxValue": "{a}*{c}*2*{pi}/{t}", "showPrecisionHint": false}], "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "prompt": "<p>Enter your answer correct to 3 decimal places.</p>\n<p>Answer = [[0]]</p>"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"name": "c", "description": "", "templateType": "randrange", "definition": "random(1..4#1)", "group": "Ungrouped variables"}, "a": {"name": "a", "description": "", "templateType": "randrange", "definition": "random(2..16#1)", "group": "Ungrouped variables"}, "t": {"name": "t", "description": "", "templateType": "randrange", "definition": "random(2..8#1)", "group": "Ungrouped variables"}, "k": {"name": "k", "description": "", "templateType": "randrange", "definition": "random(1..4#1)", "group": "Ungrouped variables"}, "k2": {"name": "k2", "description": "", "templateType": "anything", "definition": "k+c", "group": "Ungrouped variables"}}, "extensions": [], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "<p>\\(\\int\\int_R\\frac{\\var{a}}{\\var{t}\\sqrt{x^2+y^2}}dxdy\\)</p>\n<p>Limits</p>\n<p style=\"padding-left: 30px;\">\\(x^2+y^2\\ge\\simplify{{k}^2}\\) and&nbsp;\\(x^2+y^2\\le\\simplify{{k2}^2}\\).</p>\n<p style=\"padding-left: 30px;\">This is the area enclosed between two circles the first having radius \\(\\var{k}\\) and the second with radius \\(\\var{k2}\\) and both having centre (0, 0)&nbsp;</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;\">\\(\\var{k}\\le r\\le\\var{k2}\\)&nbsp; &nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp; &nbsp; \\(0\\le \\theta\\le2\\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 \\frac{\\var{a}}{\\var{t}\\sqrt{x^2+y^2}}dxdy=\\frac{\\var{a}}{\\var{t}\\sqrt{r^2}}=\\frac{\\var{a}}{\\var{t}r}\\)</p>\n<p>The Integral</p>\n<p style=\"padding-left: 30px;\">\\(\\int\\int_R\\frac{\\var{a}}{\\var{t}\\sqrt{x^2+y^2}}dxdy=\\int_0^{2\\pi}\\int_\\var{k}^{\\var{k2}}\\frac{\\var{a}}{\\var{t}r}rdrd\\theta=\\int_0^{2\\pi}\\int_\\var{k}^{\\var{k2}}\\frac{\\var{a}}{\\var{t}}drd\\theta\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Inner integral</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{k}^{\\var{k2}}\\frac{\\var{a}}{\\var{t}}dr=\\frac{\\var{a}}{\\var{t}}r\\big|_\\var{k}^{\\var{k2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{a}}{\\var{t}}(\\var{k2})-\\frac{\\var{a}}{\\var{t}}(\\var{k})\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{c}*{a}}}{\\var{t}}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Outer Integral</p>\n<p style=\"padding-left: 30px;\">\\(\\int_0^{2\\pi}\\frac{\\simplify{{c}*{a}}}{\\var{t}}d\\theta\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{c}*{a}}}{\\var{t}}\\theta\\big|_0^{2\\pi}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{c}*{a}}}{\\var{t}}({2\\pi})-0\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{c}*{a}*2*{pi}/{t}}\\)</p>", "statement": "<p>Evaluate the integral below using polar co-ordinates:</p>\n<p style=\"padding-left: 30px;\">\\(\\int\\int_R\\frac{\\var{a}}{\\var{t}\\sqrt{x^2+y^2}}dxdy\\)</p>\n<p style=\"padding-left: 30px;\">where \\(R\\) is the region of the plane enclosed by the circles \\(x^2+y^2\\ge\\simplify{{k}^2}\\) and \\(x^2+y^2\\le\\simplify{{k2}^2}\\).</p>\n<p style=\"padding-left: 30px;\"></p>", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}