// Numbas version: exam_results_page_options
{"name": "Double integral in polar coordinates - concentric circles", "extensions": ["eukleides", "geogebra", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Double integral in polar coordinates - concentric circles", "tags": [], "metadata": {"description": "<p>Double integral in polar coordinates - concentric circles</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Compute the integral&nbsp;</p>\n<p>\\[\\iint_R \\frac{y^2}{\\sqrt{x^2 + y^2}}\\, dx\\, dy\\]&nbsp;</p>\n<p>where $R$ is the region between the circles $x^2 + y^2 = \\var{r1}$ and $x^2 + y^2 = \\var{r2}$.&nbsp;</p>", "advice": "<p>Recall that ew pass to polar coordinates using $x=r\\cos(\\theta)$ and $r\\sin(\\theta)$.</p>\n<p>a) Plug the values above into $ \\frac{y^2}{\\sqrt{x^2 + y^2}}$ we get&nbsp;</p>\n<p>\\[ \\frac{r^2\\sin^2(\\theta)}{\\sqrt{r^2\\cos^2(\\theta) + r^2\\sin^2(\\theta)}} = \\frac{r^2\\sin^2(\\theta)}{r\\sqrt{\\cos^2(\\theta) + \\sin^2(\\theta)}} = r\\sin^2(\\theta).\\]</p>\n<p></p>\n<p>b) The area we are interested is the area between the concentric circles below (this shape is called and annulus)</p>\n<p>{diagram}</p>\n<p>This are can be described by the inequalities&nbsp;</p>\n<p>\\[0&lt;\\theta\\leq 2\\pi \\quad \\mbox{ and }\\quad \\var{r1}&lt;r&lt;\\var{r2}\\]&nbsp;</p>\n<p>in polar coordinates.</p>\n<p></p>\n<p>c) Note that the order the integration does not matter as boundaries for both variables are constants. The integral thus can be written as&nbsp;</p>\n<p>\\[\\int_{\\var{r1}}^{\\var{r2}}\\int_{0}^{2\\pi} r^2\\sin^2(\\theta)\\, d\\theta\\,dr.\\]</p>\n<p></p>\n<p>d)&nbsp;</p>\n<p>\\[\\int_{\\var{r1}}^{\\var{r2}}\\int_{0}^{2\\pi} r^2\\sin^2(\\theta)\\, d\\theta\\,dr = \\int_{\\var{r1}}^{\\var{r2}}r^2 \\int_{0}^{2\\pi} \\sin^2(\\theta)\\, d\\theta\\,dr&nbsp; = -\\int_{\\var{r1}}^{\\var{r2}}r^2 \\left[\\frac{\\sin(2\\theta) - 2\\theta}{4}\\right]_0^{2\\pi}\\,dr&nbsp; <br/>= \\\\[3mm ]\\int_{\\var{r1}}^{\\var{r2}}r^2 \\pi\\,dr = \\pi\\left[\\frac{r^3}{3}\\right]_{\\var{r1}}^{\\var{r2}} &nbsp;= \\pi\\left(\\frac{\\var{r2}^2 - \\var{r1}^2}{3}\\right) = \\var{ans}.\\]</p>", "rulesets": {}, "extensions": ["eukleides", "geogebra", "jsxgraph"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"r1": {"name": "r1", "group": "Ungrouped variables", "definition": "random(1 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "r2": 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