Integration in polar coordinates Worksheet by Mike May, S.J. - maymk@slu.edu Revised by Russell Blyth - blythrd@slu.edu restart:
<Text-field style="Heading 2" layout="Heading 2">A review of plotting in polar coordinates</Text-field> The first task in computing double integrals using polar coordinates is to correctly sketch graphs of functions described in polar coordinates. On a calculator you switch to polar mode. With Maple you use the coord=polar option in the plot command. You can plot the curve either with r as a function of theta, or with both r and theta described as functions of a parameter t. When we plot in polar coordinates it is usually wise to use the scaling=CONSTRAINED option so the axes have the same scale. plot(cos(2*theta),theta=0..2*Pi, coords=polar, scaling=CONSTRAINED); plot([2+sin(2*t),Pi*sin(t),t=0..Pi], coords=polar, scaling=CONSTRAINED); 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NSdlMisiISM9NyQkIjMxXD1cVlIpUSkpKiEjPSQhM1Y5KzEoKiplc3YnISM+NyQkIjMnSCUzcGA6ITQoKiohIz0kITM6KT1MVjstV1MkISM+NyQkIjJjKioqKioqKioqKioqKioqISM8JCEzYVRFW10nZWZEIiEjRC1JJ0NPTE9VUkc2IjYmSSRSR0JHNiIkIiM1ISIiJCIiISIiISQiIiEiIiEtSShTQ0FMSU5HRzYiNiNJLENPTlNUUkFJTkVERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiLUklVklFV0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSShERUZBVUxURzYiSShERUZBVUxURzYi 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 To plot several polar curves together at the same time we use set notation, just like we did in Cartesian coordinates. plot({1,2*sin(theta)}, theta=0..2*Pi, coords=polar, scaling=CONSTRAINED); NiYtSSdDVVJWRVNHNiI2JDdTNyQkIiIiIiIhJCIiISIiITckJCIzIypwI2ZWUGlqISoqISM9JCIzKzkmbyd6InlfTyIhIz03JCQiM1sjeS8tNS1RbiohIz0kIjNLIXk9ViYqKUdMRCEjPTckJCIzJzRnPz9bJ2VbIyohIz0kIjNSXUY8PC42LlEhIz03JCQiMzVENVNDNDJgJykhIz0kIjNJLCp5KEg0VTddISM9NyQkIjNGUV5aVD9EL3ohIz0kIjMlPXBEKmVjZURoISM9NyQkIjMyXj5VIT11RDMoISM9JCIzdz44MipvVSZmcSEjPTckJCIzJyllXHBoZFg7aCEjPSQiM0ApKlFqUCE+OCJ6ISM9NyQkIjNbeUhldjp4NV0hIz0kIjNaWyUzdzdFU2wpISM9NyQkIjM5ZylweW1SLCNRISM9JCIzUGkuWixiY1QjKiEjPTckJCIzOS5vSy5qUUREISM9JCIzIWUkcFNGIm9lbiohIz03JCQiM0psUyJIJm84WDghIz0kIjMjZmZJJ2Z0NjQqKiEjPTckJCEzKVJeTDhKTzUoZiEjQCQiMi02T3RAKSoqKioqKiEjPDckJCEzM0AsQUJCW2k4ISM9JCIzcztOZCNIWm4hKiohIz03JCQhMyFwKCkqZjdAPVlFISM9JCIzKXBdVSZmRGBWJyohIz03JCQhMyRSI3BYIyopM0p4JCEjPSQiM2JCQ2JebCczRSohIz03JCQhM25odCxOX0pVXSEjPSQiMzsyKj5jNCZvTicpISM9NyQkITNLXykqb1ElKltSZyEjPSQiM1JlKlJwMUktKHohIz03JCQhM2tcIlE4SjskKjMoISM9JCIzYjpzKGY0c0YwKCEjPTckJCEzVXRBPHRUPy96ISM9JCIzKHldJDRFdWtEaCEjPTckJCEzJD46Rz5yMSRmJykhIz0kIjM/UGF3ZC9rLF0hIz03JCQhMyhIOlQxRE80QiohIz0kIjNpIipSJVIoSHZYUSEjPTckJCEzRDNHaSRcQk9tKiEjPSQiMypbbzZmKVElPWQjISM9NyQkITNadzVyNSsuMioqISM9JCIzSmg1WG9dVWc4ISM9NyQkITMncClmMjQ+JioqKioqISM9JCIzXilwYik+aEosSiEjPzckJCEzOCl6VyJHIVElMyoqISM9JCEzKVs5YXF5SixOIiEjPTckJCEzOyYzdmJbbFNuKiEjPSQhM3ciKj1HZUhHS0QhIz03JCQhM2VEWExUQUVqIyohIz0kITM4Kil6c21NQW5QISM9NyQkITM1Q1ohNDwnPXUnKSEjPSQhMyIzNS4pPTN6dlwhIz03JCQhMzcpb0ouI3kxWXohIz0kITMqZWdJWz9XNzInISM9NyQkITNlJzQyZSpmYzVyISM9JCEzIz0kNHBfeE1KcSEjPTckJCEzajlVLCpcJ2VbZyEjPSQhM1dXcVpOJkdMJ3ohIz03JCQhM086OWQmXClvISpcISM9JCEzMiozJjNuTGlsJykhIz03JCQhMyZRWURxYyJ5c1AhIz0kITNcJFskUTAqKio0RSohIz03JCQhM0wnR1VGbmwnM0UhIz0kITMmUjRnN25bUGwqISM9NyQkITM9NGQsc29jIkgiISM9JCEzJGV1ai8pPkM7KiohIz03JCQhMzVlLCJ5WCRSZEkhIz8kITM7JypwaGhLJioqKioqISM9NyQkIjNuKnkhUl5KcyhHIiEjPSQhMzJeNk9lPXU7KiohIz03JCQiMy9DYU9WNy9iRCEjPSQhM14jNDlsbHohbycqISM9NyQkIjMncCc+OSZHKXpOUSEjPSQhM11fJnlRQnhdQiohIz03JCQiMyFwK10mei9JLl0hIz0kITNvIipvTW13TWUnKSEjPTckJCIzMi1HJyozLk40aCEjPSQhM0Zyb3FodCFvInohIz03JCQiM0VPXCZSSSskKjQoISM9JCEzdzVUd1ZAc1VxISM9NyQkIjNLeXRJPyR5LCF6ISM9JCEzKVwnNExZJ1EzOCchIz03JCQiM0IlUWJeWFBWbikhIz0kITNpQ0YtenFfdlwhIz03JCQiM2BibyNbITQrRCMqISM9JCEzcyl5X0JwbypmUSEjPTckJCIzWWZ3I2UkKSlvYScqISM9JCEzZENoVmpSPTBFISM9NyQkIjNtJlxeIj1iJnAhKiohIz0kITNcJyo9I2VWbjRPIiEjPTckJCIyKSkqKioqKioqKioqKioqKiohIzwkITMuVUVbXSdlZkQiISNELUknQ09MT1VSRzYiNiZJJFJHQkc2IiQiIzUhIiIkIiIhIiIhJCIiISIiIS1JJ0NVUlZFU0c2IjYkN11xNyQkIiIhIiIhJCIiISIiITckJCIzKzkmbyd6InlfTyIhIz0kIjNkREkya0R3aiQqISM/NyQkIjNsdy4jejIpKVxxIyEjPSQiM2dVenY6IXB6cyQhIz43JCQiMyJRWEctJDRKSVEhIz0kIjN5Zi5tKilcWEV3ISM+NyQkIjN2PUhNLHNJLFwhIz0kIjMrM1pAJmU1TkciISM9NyQkIjNeKjRhXiZ5KT4tJyEjPSQiMyF6SSZbLSNRbCwjISM9NyQkIjNpJlIyUSQqeVkuKCEjPSQiM1dqciJwaEhGKkchIz03JCQiM19PbjZ3IlJtI3ohIz0kIjNjJlJuc1w2TSFSISM9NyQkIjMpR3psO3dtWG4pISM9JCIzNjxyWDpGKFstJiEjPTckJCIzKilSOnJfbFhpIyohIz0kIjNubEVaQ2d6SWkhIz03JCQiMzNZKWUqKmVNT28qISM9JCIzej9zJ0gkKmZYXSghIz03JCQiM1YnSCUpUiY+YD0qKiEjPSQiMyF6J1wtLiNRaHMpISM9NyQkIjM5OzEvYnAlKioqKiohIz0kIjMua1N3JmZHdScqKiEjPTckJCIzJXAjR1hdJSk9QCoqISM9JCIzTWBjPkcvSUQ2ISM8NyQkIjNSdSZvP2VZeW4qISM9JCIzKTRkLyR5JHo8RCIhIzw3JCQiM0YjUUBIMHl0RCohIz0kIjNjIT06KSopZjt5OCEjPDckJCIzeHBYInBpcEVuKSEjPSQiM2ckZllWT1Z5XCIhIzw3JCQiMz0+OmJBaXBPeiEjPSQiMzosQFo/IVwkMzshIzw3JCQiM3I0cDIkPjkzMSghIz0kIjMoSE1mJGUxODM8ISM8NyQkIjNFayZIJGYhcCwuJyEjPSQiMy1tKSopb1NHeHoiISM8NyQkIjNzbipcJG80MSgpWyEjPSQiMzopUXouW1tDKD0hIzw3JCQiMzttIT1nZ3JSIVEhIz0kIjNOXkxZRUsjWyM+ISM8NyQkIjNqRXowKXlCZW0jISM9JCIzd1snUXA4N1EnPiEjPDckJCIzeThZO1YpPiNSOCEjPSQiMyNvc0V6KD0qNCo+ISM8NyQkITNTJ296JFxxPyU+IiEjPyQiM2MyWCRwRyoqKioqPiEjPDckJCEzX0xrNk51Um84ISM9JCIzKSkqM1IkKT4kZiEqPiEjPDckJCEzKT0vW0xoYCYqcCMhIz0kIjM1YSN5JFFHKEcnPiEjPDckJCEzLmxIYTcrVU5SISM9JCIzZ09sQVBtST4+ISM8NyQkITNpeHJFcyczUDUmISM9JCIzPUgiUlhTYSpmPSEjPDckJCEzJlstK3J5LiMqMychIz0kIjMnKUdPZls4QiR6IiEjPDckJCEzdG5RXShlXiUpKXAhIz0kIjNnXEtyJilIRjo8ISM8NyQkITM7bSc0SCgpZXojeiEjPSQiM2c2JioqekEoWzQ7ISM8NyQkITMiUksiXC8kcCgzKCkhIz0kIjMxT1VUVDZdIlwiISM8NyQkITNnZXNSVVROTSMqISM9JCIzPEdJKlEoUXYkUSIhIzw3JCQhMyEpUUNraVNBRicqISM9JCIzZHUhKlJZOFxxNyEjPDckJCEzISkzO2VnQycpNCoqISM9JCIzYVV4Q19PJ1I4IiEjPDckJCEzTWtCLl9rJykqKioqISM9JCIzK3AqZXYlKT0kWyoqISM9NyQkITN5S24oW0c/dCIqKiEjPSQiM2BpSE5tIlJucikhIz03JCQhMzBnP3UqZXRPbyohIz0kIjNdLjd3eDdyL3YhIz03JCQhMygqemAxRFp5ZCMqISM9JCIzcilbOlp4TiQ+aSEjPTckJCEzISpHZkpkeDlpJykhIz0kIjNaNTd2YDlHLl0hIz03JCQhMyVRNyZSdnk4WnohIz0kIjM5PiNcS2FjLCRSISM9NyQkITMtQiJRdzUhKSoqNCghIz0kIjNYNik+dD1qeiZIISM9NyQkITNxInA3WmRWLzQnISM9JCIzM0RNSCZRUSdvPyEjPTckJCEzTW5fYyo+bTEoXCEjPSQiMz1qLXElPndHSyIhIz03JCQhMyopKXpROC0oZmlRISM9JCIzMyEzLjUkSCo0dyghIz43JCQhM3dtTzQnUmFicCMhIz0kIjN2XCx1TEZeLFAhIz43JCQhM3k5TCJ6TlY2UiIhIz0kIjNHOzVqSVRuQigqISM/NyQkITMpXCMpejFULUU/JyEjPyQiMztoYDBOQmpCPiEjQTckJCIzZFhqYyY0J1I+OCEjPSQiM0hcNVQocFZBdSkhIz83JCQiM2x3M1BHJVJibiMhIz0kIjNWUCFHW282ZGskISM+NyQkIjN2bCIpUjNLQTpRISM9JCIzM2BtdnJ0Lmt2ISM+NyQkIjNZa1xHTFVcKipbISM9JCIzJnooNDsnUiJcI0ciISM9NyQkIjNPRVx4aW46ISpmISM9JCIzR2s0QERxaSMqPiEjPTckJCIzX1lAISl5ZE56cCEjPSQiMyMpNEtsSFhSUUchIz03JCQiM3VXKFI1SGxwKHkhIz0kIjN1J28ob3FhT1JRISM9NyQkIjN3NGAmPj0oPUsnKSEjPSQiM1JZcVxhKSlwXlwhIz03JCQiM0NIP3gjUTIsQSohIz0kIjMlPmIiSGRwTkdoISM9NyQkIjMvWGYyVE9dWycqISM9JCIzNFZ5JCpRNytzdCEjPTckJCIzPE1HanFbOi8qKiEjPSQiM1UrWT9sKil6PScpISM9NyQkIjN5Y2QqcFdzJCoqKiohIz0kIjN1alYmUUNxeikpKiEjPTckJCIzM3IuWT0pW2AiKiohIz0kIjMteEY+KGVTKUg2ISM8NyQkIjMxPS1RJEh3TGoqISM9JCIzUGYpeXMtI0hvNyEjPDckJCIzPiNROjdZRWRAKiEjPSQiMz1scSJ5JWY/KVEiISM8NyQkIjNwd0sscGBbXCcpISM9JCIzUkFNIW9jZz1dIiEjPDckJCIzdyw0OXpwQSIqeSEjPSQiMytoISk+J2ZOVWgiISM8NyQkIjMydTUwWmYleilwISM9JCIzIXpEXVxRQWByIiEjPDckJCIzKik0cSEpZVAuZWchIz0kIjMhXCJbbShSOWN6IiEjPDckJCIzKkhgKlJhMG9PXSEjPSQiM2s/cjpvcypRJz0hIzw3JCQiM10ncGdKP2lPJFEhIz0kIjMzcHJ6O2tmQj4hIzw3JCQiM0Mlb3J0I3pcaEQhIz0kIjMkNEItLjVQbSc+ISM8NyQkIjMoZjgpeilveT1LIiEjPSQiM2NOWmEhb0M3Kj4hIzw3JCQiM25KXTRPJGVaNichIz8kIjMnelshcC84KSoqKj4hIzw3JCQhMzpmLzVeeFJkNyEjPSQiMz14XmFlSzEjKj4hIzw3JCQhM3diYlo6TythRCEjPSQiM2FWdXEiUU5vJz4hIzw3JCQhM3VXLzZYJVwoeVAhIz0kIjN3Syl5WGtjZSM+ISM8NyQkITN6WHJJYCVvLyVcISM9JCIzKUguJSpbR04lcD0hIzw3JCQhMygzdUgmZTEzb2chIz0kIjNRJ0g5XlJbW3oiISM8NyQkITM/JSpIUCRveVozKCEjPSQiM3lRU0pJSXQwPCEjPDckJCEzWUYjbyRbVHZUeiEjPSQiM2N4KHAsZSlvMjshIzw3JCQhMzQ8KUdpK2pTbSkhIz0kIjNBJHBFaW9SJCpcIiEjPDckJCEzeHdPKWZdLzJEKiEjPSQiM0grZTlfYnp6OCEjPDckJCEzSHMmWy0hKjRMbiohIz0kIjNLQXYvd25eYDchIzw3JCQhMytxaVVoXC1CKiohIz0kIjMpKjMjKTM/dyRRNyIhIzw3JCQhMyc+JSp5NiopeicqKioqISM9JCIzKDMyYClRcSkqPioqISM9NyQkITNULTU6WURNOyoqISM9JCIza10uUHVhPzQoKSEjPTckJCEzI1xJND1yVnBvKiEjPSQiM0dnIXA5XU91XighIz03JCQhMzdaIT1qV14pWyMqISM9JCIzWScqKm8mKjRNdj4nISM9NyQkITMmcExULD0hKT1qKSEjPSQiMyg0bzxHJVI8XlwhIz03JCQhMyEqNHkoNEciPlF6ISM9JCIzWksqUiEpeWclPVIhIz03JCQhMyQ9YzZmJEhrQHIhIz0kIjNOUigpPWg7KCl6SCEjPTckJCEzIz05a2c1aSpIaCEjPSQiM180Xi5IOjkqNCMhIz03JCQhM1MsR1QwIlsvLiYhIz0kIjMsXipvbXAnUmQ4ISM9NyQkITMvMD00dmUiXCpRISM9JCIzYV9OVSt4KnAqeSEjPjckJCEzdVghbydwKDNtcCMhIz0kIjMvY190QVpZL1AhIz43JCQhMzVoIlwtY240TyIhIz0kIjMsQiRwVCopXFdJKiEjPzckJCEzdSRHbDRJPD5eIyEjRCQiMygqZVZXalUnWzokISNMLUknQ09MT1VSRzYiNiZJJFJHQkc2IiQiIiEiIiEkIiM1ISIiJCIiISIiIS1JKFNDQUxJTkdHNiI2I0ksQ09OU1RSQUlORURHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiItSSVWSUVXRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiRJKERFRkFVTFRHNiJJKERFRkFVTFRHNiI= Plotting several curves together allows us to plot regions made of a number of curves. The region plotted below is the portion of an annulus between two specified angles. The fifth (empty) curve is included so that Maple includes the origin in the plot (the t-range for this "curve" can be any nonempty interval). plot({[1+t,Pi/6,t=0..1],[1+t,Pi/3,t=0..1],[1,t,t=Pi/6..Pi/3], [2,t,t=Pi/6..Pi/3],[0,0,t=0..2*Pi]},coords=polar, scaling=CONSTRAINED); 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<Text-field style="Heading 2" layout="Heading 2">Exercise</Text-field> 1. Plot the curves NiMvJSJyRyomIiIkIiIiLSUkY29zRzYjJSZ0aGV0YUdGJw== and NiMvJSJyRywmIiIiRiYtJSRjb3NHNiMlJnRoZXRhR0Ym on the same graph. Find the points of intersection. (You may want to use either your calculator or paper and pencil to find the points of intersection.)
Once we can sketch curves in polar coordinates the process of setting up an integral in polar coordinates is similar to the process involved in setting up a double integral in Cartesian coordinates. The biggest challenge is finding the correct limits of integration. We will also discuss switching the order of integration.
<Text-field style="Heading 2" layout="Heading 2">Finding the limits of integration</Text-field>
<Text-field style="Heading 3" layout="Heading 3">Setting up drd<Equation executable="false" style="2D Comment" input-equation="theta">NiMlJnRoZXRhRw==</Equation> integrals</Text-field> Consider first the case of integrals using the order of integration drdNiMlJnRoZXRhRw==. Since dr is on the inside we have a region bounded by curves NiMvJSJyRy0lImdHNiMlJnRoZXRhRw== and 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 with the value of NiMlJnRoZXRhRw== bound by two constant angles. The integration with respect to r for a particular NiMlJnRoZXRhRw== is along a radial line. The following block of code is designed to help you visualize what the limits of integration mean. The curves NiMvJSJyRy0lImdHNiMlJnRoZXRhRw== and 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 are specified in the fourth and fifth lines as lowr and highr (the inside and outside curves, respectively). r:='r': theta := 'theta': lowtheta := Pi/6; hightheta := 3*Pi/2; lowr := theta -> 1.5-sin(theta); highr := theta -> 3 + cos(theta); print(`Region of integration for `, Int(Int(f(r, theta)*r,r=lowr(theta)..highr(theta)), theta=lowtheta..hightheta)); inside := plot([lowr(theta), theta,theta=lowtheta..hightheta], color=red, coords=polar): outside := plot([highr(theta), theta,theta=lowtheta..hightheta], color=green, coords=polar): line := {}: for i from 0 to 10 do tval := evalf(lowtheta + i/10*(hightheta-lowtheta)): if (abs(evalf(lowr(tval) - highr(tval)))>0) then line := line union {[r,tval, r=evalf(lowr(tval))..evalf(highr(tval))]}; end if end do: plotlines := plot(line,coords=polar, color=BLACK) : plots[display]({inside, outside, plotlines},scaling=CONSTRAINED); LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIic= LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiQiIiM= Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmJCIjOiEiIiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjOSRGLEYlRiVGJQ== Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmIiIkIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYrRiVGJUYl 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Note that since we integrate with respect to r first, the r-limits are functions of theta while the theta limits are constants. We first integrate from the inside (red) curve to the outside (green) curve, along the radial lines, and then add up (integrate) over the range of values of theta.
<Text-field style="Heading 2" layout="Heading 2">Exercises</Text-field> 2. Find the limits of integration to integrate over the region inside the curve NiMvJSJyRywmIiIiRiYtJSRjb3NHNiMlJnRoZXRhR0Ym and outside the curve r=1. Modify the code above to show that you have the correct region. 3. Find the limits of integration to integrate over the region inside the curve NiMvJSJyRyomIiIkIiIiLSUkY29zRzYjJSZ0aGV0YUdGJw== and outside the curve NiMvJSJyRywmIiIiRiYtJSRjb3NHNiMlJnRoZXRhR0Ym. Modify the code above to show that you have the correct region.
<Text-field style="Heading 3" layout="Heading 3">Setting up d<Equation executable="false" style="2D Comment" input-equation="theta">NiMlJnRoZXRhRw==</Equation>dr integrals</Text-field> Similarly we can set up integrals using the order of integration drdNiMlJnRoZXRhRw==. Now the region of integration is bounded by curves NiMvJSZ0aGV0YUctJSJnRzYjJSJyRw== and NiMvJSZ0aGV0YUctJSJoRzYjJSJyRw== with the value of r being bound by two constants. Instead of integrating first on radial lines, we start by integrating along circular arcs with a fixed values of r. The curves NiMvJSZ0aGV0YUctJSJnRzYjJSJyRw== and NiMvJSZ0aGV0YUctJSJoRzYjJSJyRw== appear in the fourth and fifth lines of the code as lowtheta and hightheta. Remember that angles increase in a counterclockwise direction, so lowtheta is the angle on the clockwise extreme of the region. You need to take care that lowtheta is a smaller angle than hightheta (sometimes this means that lowtheta will be a negative angle). r:='r': theta := 'theta': lowr := 1; highr := 5; lowtheta := r -> Pi*r/6; hightheta := r -> Pi*(2-r/12); print(`Region of integration for `, Int(Int(f(r, theta)*r, theta=lowtheta(r)..hightheta(r)),r=lowr..highr)); lowthetacurve := plot ([r,lowtheta(r), r=lowr..highr], color=red, coords=polar) : highthetacurve := plot ([r, hightheta(r), r=lowr..highr], color=green, coords=polar) : arcs := {} : for i from 0 to 10 do tempr := evalf(lowr + i/10*(highr-lowr)): if (abs(evalf(lowtheta(tempr)-hightheta(tempr)))>0) then arcs := arcs union {[tempr, theta, theta=lowtheta(tempr)..hightheta(tempr)]}: end if: end do: grapharcs := plot(arcs,coords=polar, color=BLACK) : plots[display] ( {lowthetacurve, highthetacurve, grapharcs } ,scaling=CONSTRAINED) ; IiIi IiIm Zio2I0kickc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiOSRGLSNGLSIiJ0YlRiVGJQ== Zio2I0kickc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZJI1BpRyUqcHJvdGVjdGVkRyIiIiwmIiIjRiw5JCMhIiIiIzdGLEYlRiVGJQ== NiRJO1JlZ2lvbn5vZn5pbnRlZ3JhdGlvbn5mb3J+RzYiLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQtRiY2JComLUkiZkdGJDYkSSJyR0YkSSZ0aGV0YUdGJCIiIkYxRjMvRjI7LCQqJkkjUGlHRihGM0YxRjMjRjMiIicqJkY4RjMsJiIiI0YzRjEjISIiIiM3RjMvRjE7RjMiIiY= 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