Integration in polar coordinates Worksheet by Mike May, S.J.- maymk@slu.edu restart:with(plots): Start by opening the section below.
<Text-field style="_pstyle256" layout="_pstyle256">A review of plotting in polar coordinates:</Text-field> The first problem in trying to do double integrals in polar coordinates is to be able to sketch graphs of functions described in polar coordinates. On your calculators you switch to polar mode. With Maple you use the coords=polar option on the plot command. You can either 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 probably 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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NictJSlCT1VORFNfWEc2IyQiJCE+ISIiLSUpQk9VTkRTX1lHNiMkIiRxIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJXFPISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJWdPISIiLSUpQ0hJTERSRU5HNiI= 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To plot several 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); 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 Plotting several curves together lets us plot regions made of a number of curves. The region plotted below is the portion of an annulus between two specified angles. 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..1]},coords=polar, scaling=CONSTRAINED); 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We can also use the implicitplot command from the plots package when we want to plot a number of curves, some in terms of r and some in terms of theta. implicitplot([r=1, r=2, theta=Pi/6, theta=2*Pi/3, r*cos(theta)=1.5], r=0..3, theta=0..2*Pi, color=[red, green, blue, yellow, brown], coords=polar); 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Exercise: 1) Plot the curves 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNigtSSNtbkdGJDYkUSIxRidGPi1GOzYtUSIrRidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmhuLUYjNictRiw2JVEkY29zRicvRjVGQkY+LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Zkby1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKCZ0aGV0YTtGJ0Zfb0Y+LyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y+Rj5GXnBGPkYrRl5wRj5GK0ZecEY+RitGXnBGPg==. Find the points of intersection. (You may want to use either your calculator or a piece of paper to find the points of intersection.)
Once we can sketch curves, the problems involved in setting up an integral in polar coordinates are similar to the problems involved in setting up a double integral in Cartesian coordinates. The biggest problem is finding the correct limits of integration. We will also be concerned with switching the order of integration.
<Text-field style="Heading 2" layout="Heading 2"><Font bold="true" style="_cstyle256" size="14">Finding the limits of integration: </Font> </Text-field>
<Text-field style="Heading 3" layout="Heading 3">Setting up drd<Equation executable="false" style="2D Comment" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI=</Equation> integrals</Text-field> For our first example we want to consider the region bounded by the curves theta=Pi/3, theta=3*Pi/2, r=1.5-sin(theta), and r=3+cos(theta). As with the Cartesian case, we start by plotting the curves implicitly to see what is going on. implicitplot([r=1.5-sin(theta), r=3+cos(theta), theta=Pi/6, theta=3*Pi/2], r=0..4, theta=0..2*Pi, coords=polar, axes=boxed, color=[red, green, blue, yellow]); 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Given the picture, we have a region bounded by curves 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYjNiYtRiw2JVEiaEYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSgmdGhldGE7RicvRjVGQkY+Rj5GPkY+RitGPkYrRj4= with the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== bound by two constants. Thus we want to integrate in r first, using drdLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==. We check that the integral sets up correctly. 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)); LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIic= LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiQiIiM= Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmJCIjOiEiIiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjOSRGLEYlRiVGJQ== Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmIiIkIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYrRiVGJUYl NiRJO1JlZ2lvbn5vZn5pbnRlZ3JhdGlvbn5mb3J+RzYiLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQtRiY2JComLUkiZkdGJDYkSSJyR0YkSSZ0aGV0YUdGJCIiIkYxRjMvRjE7LCYkIiM6ISIiRjMtSSRzaW5HRic2I0YyRjksJiIiJEYzLUkkY29zR0YnRjxGMy9GMjssJEkjUGlHRigjRjMiIicsJEZEI0Y+IiIj The integration with respect to r for a particular LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI= integrates along a radial line. The following block of code is designed to help you visualize what the limits of integration mean. r:='r': theta := 'theta': lowtheta := Pi/3; hightheta := 3*Pi/2; lowr := theta -> 1; 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); LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIiQ= LCRJI1BpRyUqcHJvdGVjdGVkRyMiIiQiIiM= Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSIiIkYlRiVGJQ== Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmIiIkIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYrRiVGJUYl NiRJO1JlZ2lvbn5vZn5pbnRlZ3JhdGlvbn5mb3J+RzYiLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQtRiY2JComLUkiZkdGJDYkSSJyR0YkSSZ0aGV0YUdGJCIiIkYxRjMvRjE7RjMsJiIiJEYzLUkkY29zR0YnNiNGMkYzL0YyOywkSSNQaUdGKCNGM0Y3LCRGPiNGNyIiIw== 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