Polar Equations and Area bounded by Polar CurvesCalculus III Lab on Section 11.3Restart and load the plots library:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=
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...</Font></Text-field>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 ...These are called rose curves. You may have seen these in the design of rose windows in churches. 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and 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 are n-leaved roses when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is an odd integer greater than 1 and are 2n-leaved roses when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== is an even integer. Note the range of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 needed to generate the curves when n is odd verses when n is even. 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
<Text-field superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="12" bold="false" subscript="false" family="Times New Roman" opaque="false" underline="false" background="[255,255,255]" readonly="false" foreground="[0,0,0]" layout="Heading 2"><Equation executable="false" input-equation="" display="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">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</Equation></Text-field>The graph of this equation is called a limacon. There are four different types:Limacon with inner loop: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 (oval) 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JSFH
<Text-field style="Heading 1" layout="Heading 1">Calculating the Area in a Region Bounded by a Polar Curve</Text-field> Calculating the area bounded by a polar curve . The area bounded by 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 between the limits of the lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYlUSgmYWxwaGE7RidGNEY3RjdGK0Y3 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmdGhldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYlUScmYmV0YTtGJ0Y0RjdGN0YrRjc= is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiwtSShtc3Vic3VwR0YkNictRjs2LVEmJmludDtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tRiw2JVEoJmFscGhhO0YnL0Y1RkJGPi1GLDYlUScmYmV0YTtGJ0Zcb0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnRj4tRiM2JS1GX3A2JEZib0Y+RjRGNy8lLmxpbmV0aGlja25lc3NHRmFwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmpwLyUpYmV2ZWxsZWRHRkJGPkY+LUY7Ni1RIn5GJ0Y+RkBGQ0ZFRkdGSUZLRk1GZm5GaG4tRiw2JVEiZkYnRjRGNy1JJW1zdXBHRiQ2JS1GZ282JC1GIzYkLUYsNiVRKCZ0aGV0YTtGJ0Zcb0Y+Rj5GPkZkcC9GYW9GZW9GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zlci8lKmxpbmVicmVha0dRJWF1dG9GJy1GOzYtUTAmRGlmZmVyZW50aWFsRDtGJ0Y+RkBGQ0ZFRkdGSUZLRk1GZm5GaG5GXHJGPkYrRj5GK0Y+. Important Note: The lower limit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk0NTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 must be less than the upper limit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk0NjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3. Example: Find the area of one petal of the rose 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 . We plot the curve: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We need to find the limits of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== that generate one petal. It can help to determine the value(s0 of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3 for which r = 0 -- This is easy to do by hand -- but we can also ask Maple to do this: 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 only gives us on of the values -- to find all we should use the option "allsolutions=true") and see that all solutions 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, where n is any integer: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 To get the right petal, we can choose the values for the limits 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. We confirm this in plotting over these limits below:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEoYW5pbWF0ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYtLUYsNiVRKnBvbGFycGxvdEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2Ji1GIzYyLUkjbW5HRiQ2JFEiM0YnRkEtRj42LVEifkYnRkFGQy9GR0ZFRkhGSkZMRk5GUEZSL0ZWRlQtRiw2JVEkY29zRicvRjBGRUZBLUY2NiQtRiM2JkZmbi1GPjY+UTEmSW52aXNpYmxlVGltZXM7RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxOEYnLyUlYm9sZEdGRUZiby8lKnVuZGVybGluZUdGRS8lKnN1YnNjcmlwdEdGRS8lLHN1cGVyc2NyaXB0R0ZFLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkUvJStleGVjdXRhYmxlR0ZFLyUpcmVhZG9ubHlHRkUvJSljb21wb3NlZEdGRS8lKmNvbnZlcnRlZEdGRS8lK2ltc2VsZWN0ZWRHRkUvJSxwbGFjZWhvbGRlckdGRS8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGRUZBRkNGXW9GSEZKRkxGTkZQRlJGXm8tRiw2JVEoJnRoZXRhO0YnRmJvRkFGQUZBRj1GXnItRj42LVEiPUYnRkFGQ0Zdb0ZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGZXItRj42LVEqJnVtaW51czA7RidGQUZDRl1vRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJy9GVkZbcy1JJm1mcmFjR0YkNigtRiw2JVElJnBpO0YnRmJvRkEtRiM2JC1GZ242JFEiNkYnRkFGQS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXQvJSliZXZlbGxlZEdGRS1GPjYtUSMuLkYnRkFGQ0Zdb0ZIRkpGTEZORlBGanJGXm8tRl5zNigtRiM2Jy1GLDYlUSJBRidGL0YyLUY+Ni1RJyZzZG90O0YnRkFGQ0Zdb0ZIRkpGTEZORlBGUkZeb0ZqbkZgc0ZBRmNzRmhzRlt0Rl50RmB0Rj0tRiw2JVEoc2NhbGluZ0YnRi9GMkZhci1GLDYlUSxDT05TVFJBSU5FREYnRi9GMkZBRkEvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPUZpdEZhckZnci1GZ242JEZqc0ZBRmJ0Rlt2RkFGQS1GLDYjUSFGJ0ZgcUZBComputing the integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1GLzYkUSIyRidGMi8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjwvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0YzUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHRjEvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGSC8lKWJldmVsbGVkR1EmZmFsc2VGJy1JI21vR0YkNi1RJyZzZG90O0YnRjIvJSZmZW5jZUdGTS8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zcby1JKG1zdWJzdXBHRiQ2Jy1GTzYtUSsmSW50ZWdyYWw7RidGMi9GU1EmdW5zZXRGJy9GVUZmby9GV0Y8L0ZZRmZvL0ZlbkY8L0ZnbkZmby9GaW5GZm9Gam5GXW8tRiM2Ky1GTzYtUSomdW1pbnVzMDtGJ0YyRlJGVEZWRlhGWkZmbkZobi9GW29RLDAuMjIyMjIyMmVtRicvRl5vRmNwLUYjNictSSNtaUdGJDYjUSFGJy1GLDYoLUYjNiotRmhwNiZRJyYjOTYwO0YnL0Y7Rk0vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjxGMkY6LyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJ0Y9LyUscGxhY2Vob2xkZXJHRjxGY3FGP0ZCLUYjNiktRi82JFEiNkYnRjJGOi9GZnFRLFsyMDAsMCwyMDBdRidGPUZocUY/RkJGREZGRklGS0Y9Rj9GMkZncEY6Rl9yRj1GaHFGP0ZCLUYjNitGZ3AtRiM2J0ZncC1GLDYoLUYjNiktRmhwNiVGYXFGYnFGMkY6Rl9yRj1GaHFGP0ZCRmpxRkRGRkZJRktGPUY/RjJGZ3BGOkZfckY9RmhxRj9GQi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGXXMtSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2KS1GLzYkUSIzRidGMkZOLUZocDYlUSRjb3NGJ0ZicUYyLUZkczYkLUYjNixGaHMtRk82LVEifkYnRjJGUkZURlZGWEZaRmZuRmhuRmpuRl1vLUZocDYlUScmIzk1MjtGJ0ZicUYyRjpGZXFGPUZocUZjcUY/RkJGMkZidEY/RjJGMkY1RltzLUZPNi1RMCZEaWZmZXJlbnRpYWxEO0YnRjJGZW9GZ28vRldGZm9GaW8vRmVuRmZvRltwRlxwRmpuRl1vRmV0Rj9GMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JUYrLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGK0YxRjQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=, we have the area is 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, as seen below: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMUY0JSFH
<Text-field style="Heading 1" layout="Heading 1">Calculating the Points of Intersection of Polar Graphs.</Text-field>Points of Intersection of Polar Graphs. Because a point may be expressed in different ways in polar coordinates, points of intersection of two polar graphs may be overlooked in traditional simultaneous solving of the two equations represented in the graphs. Consider the example the intersection of the two curves of Example 5 page 650: 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We can see from above that we should find three points of intersection. However solving simultaneously (either by hand or with Maple) misses one of these solutions -- LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDY+UTAmQXBwbHlGdW5jdGlvbjtGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzE4RicvJSVib2xkR1EmZmFsc2VGJy9GN0ZILyUqdW5kZXJsaW5lR0ZILyUqc3Vic2NyaXB0R0ZILyUsc3VwZXJzY3JpcHRHRkgvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGSC8lK2V4ZWN1dGFibGVHRkgvJSlyZWFkb25seUdGSC8lKWNvbXBvc2VkR0ZILyUqY29udmVydGVkR0ZILyUraW1zZWxlY3RlZEdGSC8lLHBsYWNlaG9sZGVyR0ZILyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZIL0Y6USdub3JtYWxGJy8lJmZlbmNlR0ZILyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmJwLUkobWZlbmNlZEdGJDYkLUYjNictRmZwNiYtRiM2MS1GLDYlUSJyRidGNkY5LUY9Ni1RIj1GJ0Zgb0Zib0Zkb0Zmb0Zob0Zqb0ZccEZecC9GYXBRLDAuMjc3Nzc3OGVtRicvRmRwRmVxLUkjbW5HRiQ2JFEiMUYnRmBvLUY9Ni1RIitGJ0Zgb0Zib0Zkb0Zmb0Zob0Zqb0ZccEZecC9GYXBRLDAuMjIyMjIyMmVtRicvRmRwRl9yLUY9Ni1RJGNvc0YnRmBvL0Zjb1EmdW5zZXRGJy9GZW9GZXIvRmdvRmVyL0Zpb0Zlci9GW3BGZXIvRl1wRmVyL0ZfcEZlckZgcEZjcC1GZnA2JC1GIzYoLUYsNiZRJyYjOTUyO0YnRkkvRl9vRjhGYG9GNi9GUVErWzAsMTYwLDgwXUYnL0Zdb0Y4RmNzRjlGYG8tRj02LVEiLEYnRmBvRmJvL0Zlb0Y4RmZvRmhvRmpvRlxwRl5wRmBwL0ZkcFEsMC4zMzMzMzMzZW1GJ0ZecUZhcS1GaHE2JFEiM0YnRmBvLUY9Ni1RJyZzZG90O0YnRmBvRmJvRmRvRmZvRmhvRmpvRlxwRl5wRmBwRmNwLUYjNidGK0ZhckY8RlxzRmBvRitGWEZgb0Zgby8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGZ3MtRmZwNiYtRiM2J0ZecUZncy1GLDYlRmJzRklGYG9GWEZgb0Zgb0ZldEZodEZYRmBvRmBvRlhGYG9GK0ZYRmBvRitGWEZgbw==What point was missed in solving simultaneously for r and theta? We can see this by animating each curve separately: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Finding the Area of a Region Between Two Curves</Text-field>Example 4 page 650 Finding the Area of a Region Between Two Curves -- inside the cardiod and outside of the circle (blue in area text). Plot the cardiod, 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, and the circle, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= on the same graph: 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Find the first point of intersection: 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 Using symmetry, we calculate the area of the top blue shaded part, then double. 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Multiply by 2:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JI21pR0YkNiVRIkFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRicvJStleGVjdXRhYmxlR0Y4Ri8=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Caution on Calculating the Points of Intersection of Polar Graphs.</Text-field>Points of Intersection of Polar Graphs. Because a point may be expressed in different ways in polar coordinates, points of intersection of two polar graphs may be overlooked in traditional simultaneous solving of the two equations represented in the graphs. Consider the example the intersection of the two curves of Example 5 page 650: 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We can see from above that we should find three points of intersection. However solving simultaneously (either by hand or with Maple) misses one of these solutions -- 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What point was missed in solving simultaneously for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjdGK0Y3? We can see this by animating each curve separately: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The point missed is at the pole (origin) -- The cardiod is at the pole when theta = ___________ and the circle is at the pole when theta = __________. JSFH