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{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 17 "Double Integrals " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 221 "IMP ORTANT: This worksheet depends on some programs we have written in Ma ple. You have to execute these first. Click on the + in the box belo w, then follow the directions you see at the beginning of the hidden s ection. " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 257 29 "Programs fo r double integrals" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 287 "This sectio n contains Maple programs we will use. Place the cursor on the first \+ red line below and hit return. When Maple is done processing, scroll b ack up to the line which reads \"Programs for double integrals\" close the section by clicking on the minus sign in the gray box to left.\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 605 "with(plottools): with( plots):\nsetwindow:=proc(xmin,xmax,ymin,ymax) global win; \nif (nargs \+ =0) then win:='win': else win:=plot( 1000, x=xmin..xmax, y=ymin..ymax) : fi; end:\nplt:=proc(f,g,var,a,b)\n if (evalf(a)<>evalf(b)) then plo t([f,g,var=a..b]); fi; end:\npolarpltr:=proc(f,g,var,a,b)\n if (evalf (a)<>evalf(b)) then polarplot([f,g,var=a..b],scaling=CONSTRAINED); fi; end:\npolarplta:=proc(f,var,a,b)\n if (evalf(a)<>evalf(b)) then pola rplot(f,var=a..b,scaling=CONSTRAINED); fi; end:\npolarpltab:=proc(f,va r,a,b)\n if (evalf(a)<>evalf(b)) then polarplot(f,var=a..b,scaling=CO NSTRAINED,color=black); fi; end:\n" }{TEXT -1 12 "dxdy command" } {MPLTEXT 1 0 1342 "\ndxdy:=proc(yrange,xrange) local a,b,f,g,h,Ls,i,PP 1,PP2,PPtop,PPbot; \nif (nargs <> 2 ) then\nRETURN(print(\"Give TWO in tegration ranges.\")); fi;\nif (type(yrange,`=`) and (op(1,yrange)=y) \+ and\n type(op(2,yrange),range)\n and type(xrange,`=`) and (op (1,xrange)=x) and\n type(op(2,xrange),range)) then\n a:=op( 1,op(2,yrange));\n b:=op(2,op(2,yrange));\n f:=op(1,op(2 ,xrange));\n g:=op(2,op(2,xrange));\n else\nRETURN(print(\" You need to give 2 integration ranges, separated by a comma: y=a..b, x =ex1..ex2 so that the double integral would have the form\", Int(Int(F (x,y),x=ex1..ex2),y=a..b))); fi;\nif not(type(a,constant)) then\nRETUR N(\"Lower outside limit must be a number.\"); fi;\nif not(type(b,const ant)) then\nRETURN(\"Upper outside limit must be a number.\"); fi;\nif member(x,indets(f)) then\nRETURN( \"Lower inside limit must be in ter ms of y only.\"); fi; \nif member(x,indets(g)) then\nRETURN( \"Upper \+ inside limit must be in terms of y only.\"); fi; \nh:=(b-a)/20;\nprin t(Int(Int(F(x,y),x=f..g),y=a..b));\n PP1:=plt(f,y,y,a,b); \nPP2:=plt(g ,y,y,a,b); \nPPtop:=plt(x,b,x,subs(y=b,f),subs(y=b,g));\nPPbot:=\nplt( x,a,x,subs(y=a,f),subs(y=a,g)); Ls:=[seq(line([subs(y=a+i*h,f), a+i*h] ,[subs(y=a+i*h,g),a+i*h]),i=1..19)];\nif (win='win') then display(Ls,P P1,PP2,PPtop,PPbot); else display(win,Ls,PP1,PP2,PPtop,PPbot); fi; end :\n" }{TEXT -1 12 "dydx command" }{MPLTEXT 1 0 1355 "\ndydx:=proc(xran ge,yrange) local a,b,f,g,h,Ls,i,PP1,PP2,PPleft,PPright; \nif (nargs <> 2) then\nRETURN(print(\"Give TWO integration ranges.\")); fi;\nif (ty pe(yrange,`=`) and (op(1,yrange)=y) and\n type(op(2,yrange),range) \n and type(xrange,`=`) and (op(1,xrange)=x) and\n type(op(2, xrange),range)) then\n f:=op(1,op(2,yrange));\n g:=op(2, op(2,yrange));\n a:=op(1,op(2,xrange));\n b:=op(2,op(2,x range));\n else\nRETURN(print(\"You need to give 2 integration ran ges, separated by a comma: x=a..b, y=ex1..ex2 so that the double integ ral would have the form\", Int(Int(F(x,y),y=ex1..ex2),x=a..b))); fi;\n if not(type(a,constant)) then\nRETURN(\"Lower outside limit must be a \+ number.\"); fi;\nif not(type(b,constant)) then\nRETURN(\"Upper outside limit must be a number.\"); fi;\nif member(y,indets(f)) then\nRETURN( \"Lower inside limit must be in terms of x only.\"); fi; \nif member (y,indets(g)) then\nRETURN( \"Upper inside limit must be in terms of x only.\"); fi; \nh:=(b-a)/20;\nprint(Int(Int(F(x,y),y=f..g),x=a..b)); \n PP1:=plt(x,f,x,a,b); \nPP2:=plt(x,g,x,a,b); \nPPright:=plt(b,y,y,su bs(x=b,f),subs(x=b,g));\nPPleft:=\nplt(a,y,y,subs(x=a,f),subs(x=a,g)); Ls:=[seq(line([a+i*h, subs(x=a+i*h,f)], [a+i*h, subs(x=a+i*h,g)]),i=1 ..19)];\nif (win='win') then display(Ls,PP1,PP2,PPleft,PPright); else \+ display(win,Ls,PP1,PP2,PPleft,PPright); fi; end:\n" }{TEXT -1 16 "drdt heta command" }{MPLTEXT 1 0 1606 "\ndrdtheta:=proc(thetarange,rrange) \+ local a,b,f,g,h,Ls,i,PP1,PP2,PPtop,PPbot,pref; \nif (nargs <> 2 ) then \nRETURN(print(\"Give TWO integration ranges.\")); fi;\nif (type(theta range,`=`) and (op(1,thetarange)=theta) and\n type(op(2,thetarange ),range)\n and type(rrange,`=`) and (op(1,rrange)=r) and\n ty pe(op(2,rrange),range)) then\n a:=op(1,op(2,thetarange));\n \+ b:=op(2,op(2,thetarange));\n f:=op(1,op(2,rrange));\n \+ g:=op(2,op(2,rrange));\n else\nRETURN(print(\"You need to give 2 integration ranges, separated by a comma: theta=theta[1]..theta[2], r =ex1..ex2 so that the double integral would have the form\", Int(Int(F (r,theta)*r,r=ex1..ex2),theta=theta[1]..theta[2]))); fi;\nif is(whatty pe(evalf(a))<>float) then\nRETURN(\"Lower outside limit must be a numb er.\"); fi;\nif is(whattype(evalf(b))<>float) then\nRETURN(\"Upper out side limit must be a number.\"); fi;\nif member(r,indets(f)) then\nRET URN( \" Lower inside limit must be in terms of theta only.\"); fi; \n if member(r,indets(g)) then\nRETURN( \"Upper inside limit must be in t erms of theta only.\"); fi; \nh:=(b-a)/20;\nprint(Int(Int(F(r,theta)* r ,r=f..g),theta=a..b));\n PP1:=polarplta(f, theta,a,b); \nPP2:=polarp lta(g,theta,a,b); \nPPtop:=polarpltr(r,b,r,subs(theta=b,f),subs(theta= b,g));\nPPbot:=\npolarpltr(r,a,r,subs(theta=a,f),subs(theta=a,g)); Ls: =[seq(line([subs(theta=a+i*h,f)*cos(a+i*h), subs(theta=a+i*h,f)*sin(a+ i*h)],[subs(theta=a+i*h,g)*cos(a+i*h),\nsubs(theta=a+i*h,g)*sin(a+i*h) ]),i=1..19)];\npref:=polarplot(1,t=0..2*Pi,color=white, scaling=constr ained);\n display(PP1,PP2,PPtop,PPbot,pref,Ls,scaling=CONSTRAINED); en d:\n" }{TEXT -1 16 "dthetadr command" }{MPLTEXT 1 0 1473 "\ndthetadr:= proc(rrange,thetarange) local a,b,f,g,h,Ls,i,PP1,PP2,PPtop,PPbot,pref; \nif (nargs <> 2 ) then\nRETURN(print(\"Give TWO integration ranges. \")); fi;\nif (type(thetarange,`=`) and (op(1,thetarange)=theta) and\n type(op(2,thetarange),range)\n and type(rrange,`=`) and (op( 1,rrange)=r) and\n type(op(2,rrange),range)) then\n f:=op(1 ,op(2,thetarange));\n g:=op(2,op(2,thetarange));\n a:=op (1,op(2,rrange));\n b:=op(2,op(2,rrange));\n else\nRETURN(p rint(\"You need to give 2 integration ranges, separated by a comma: r= a..b, theta=ex1..ex2 so that the double integral would have the form\" , Int(Int(F(r,theta)*r,theta=ex1..ex2),r=a..b))); fi;\nif is(whattype( evalf(a))<>float) then\nRETURN(\"Lower outside limit must be a number. \"); fi;\nif is(whattype(evalf(b))<>float) then\nRETURN(\"Upper outsid e limit must be a number.\"); fi;\nif member(theta,indets(f)) then\nRE TURN( \" Lower inside limit must be in terms of r only.\"); fi; \nif \+ member(theta,indets(g)) then\nRETURN( \"Upper inside limit must be in \+ terms of r only.\"); fi; \nh:=(b-a)/20;\nprint(Int(Int(F(r,theta)*r,t heta=f..g),r=a..b));\n PP1:=polarpltr(r,f,r,a,b); \nPP2:=polarpltr(r,g ,r,a,b); \nPPtop:=\npolarpltr(a,t,t,subs(r=a,f),subs(r=a,g));PPbot:=\n polarpltr(b,t,t,subs(r=b,f),subs(r=b,g)); Ls:=[seq(polarpltab(a+i*h,t, subs(r=a+i*h,f),subs(r=a+i*h,g)),\n i=1..19)];\npref:=polarplot(1,t=0. .2*Pi,color=white, scaling=constrained);\n display(PP1,PP2,PPtop,PPbot ,pref,Ls,scaling=CONSTRAINED); end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " Scroll to top and close this section!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 32 "Part I, Rectangular Coordinates" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 327 "Maple can calculate double integrals as iterated integrals, eithe r numerically or by repeated use of the Fundamental Theorem of Calculu s. The tricky part is providing the appropriate limits of integration , just as it is when doing such integrals by hand. For example, if R is the region, shown below, between the graphs of " }{XPPEDIT 18 0 " y = x;" "6#/%\"yG%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = x^2; " "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "f(x,y) \+ = (x+y)^3;" "6#/-%\"fG6$%\"xG%\"yG*$,&F'\"\"\"F(F+\"\"$" }{TEXT -1 45 ", we compute the integral of f over R as " }{XPPEDIT 18 0 "Int(In t(f(x,y),y = x^2 .. x),x = 0 .. 1);" "6#-%$IntG6$-F$6$-%\"fG6$%\"xG%\" yG/F,;*$F+\"\"#F+/F+;\"\"!\"\"\"" }{TEXT -1 12 " as follows." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(int((x+y)^3,y = x^2 .. x ),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$4\"\"$/&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " \+ Region between y=x and " }{XPPEDIT 18 0 "y = x^2;" "6 #/%\"yG*$%\"xG\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 238 220 220 {PLOTDATA 2 "6;-%'CURVESG6#7$7$$\"+++++]!#6F(7$F($\"+++++D!#7-F$6#7$7$ $\"+++++5!#5F37$F3$F4F*-F$6#7$7$$\"+++++:F5F<7$F<$\"++++]AF*-F$6#7$7$$ \"+++++?F5FE7$FE$\"+++++SF*-F$6#7$7$$F-F5FN7$FN$\"++++]iF*-F$6#7$7$$\" +++++IF5FV7$FV$\"+++++!*F*-F$6#7$7$$\"+++++NF5Fin7$Fin$\"++++D7F5-F$6# 7$7$$FIF5Fbo7$Fbo$\"+++++;F5-F$6#7$7$$\"+++++XF5Fjo7$Fjo$\"++++D?F5-F$ 6#7$7$$F)F5Fcp7$FcpFN-F$6#7$7$$\"+++++bF5Fip7$Fip$\"++++DIF5-F$6#7$7$$ \"+++++gF5Fbq7$Fbq$\"+++++OF5-F$6#7$7$$\"+++++lF5F[r7$F[r$\"++++DUF5-F $6#7$7$$\"+++++qF5Fdr7$Fdr$\"+++++\\F5-F$6#7$7$$\"+++++vF5F]s7$F]s$\"+ +++DcF5-F$6#7$7$$\"+++++!)F5Ffs7$Ffs$\"+++++kF5-F$6#7$7$$\"+++++&)F5F_ t7$F_t$\"++++DsF5-F$6#7$7$$FZF5Fht7$Fht$\"+++++\")F5-F$6#7$7$$\"+++++& *F5F`u7$F`u$\"++++D!*F5-F$6$7S7$\"\"!Fiu7$$\"1nmm;arz@!#Y2aFjvFhz7$$\"1mm;zXu9cFjvF[[l7$$\"1+++]y ))GeFjvF^[l7$$\"1****\\i_QQgFjvFa[l7$$\"1***\\7y%3TiFjvFd[l7$$\"1**** \\P![hY'FjvFg[l7$$\"1LLL$Qx$omFjvFj[l7$$\"1+++v.I%)oFjvF]\\l7$$\"1mm\" zpe*zqFjvF`\\l7$$\"1+++D\\'QH(FjvFc\\l7$$\"1KLe9S8&\\(FjvFf\\l7$$\"1** *\\i?=bq(FjvFi\\l7$$\"1LLL3s?6zFjvF\\]l7$$\"1++DJXaE\")FjvF_]l7$$\"1nm mm*RRL)FjvFb]l7$$\"1mm;a<.Y&)FjvFe]l7$$\"1LLe9tOc()FjvFh]l7$$\"1+++]Qk \\*)FjvF[^l7$$\"1LL$3dg6<*FjvF^^l7$$\"1mmmmxGp$*FjvFa^l7$$\"1++D\"oK0e *FjvFd^l7$$\"1++v=5s#y*FjvFg^l7$$\"\"\"FiuFj^l-%'COLOURG6&%$RGBG$\"#5! \"\"FiuFiu-F$6$7SFhu7$F[v$\"1SVl(Hf6v%!#>7$F_v$\"1F*)>\"4,;m\"!#=7$Fbv $\"1'>E!RyNbQF]`l7$Fev$\"1rx!4%pb#)pF]`l7$Fhv$\"1hGm!pE55\"F]v7$F\\w$ \"1ps+Qo-c:F]v7$F_w$\"1(>G)30()4@F]v7$Fbw$\"1m(3M!3=rFF]v7$Few$\"1B:4y /&*>NF]v7$Fhw$\"1/`)3*\\Y$Q%F]v7$F[x$\"1wqo66VA_F]v7$F^x$\"10w6GDvaiF] v7$Fax$\"1Tciey#\\Q(F]v7$Fdx$\"101&36?Gc)F]v7$Fgx$\"1%\\-a\"[$zq*F]v7$ Fjx$\"1;-T#*)3j6\"Fjv7$F]y$\"18m5lo_Z7Fjv7$F`y$\"11AN-iL49Fjv7$Fcy$\"1 r#e.m%zg:Fjv7$Ffy$\"1NKN#))fet\"Fjv7$Fiy$\"1Y(QQm57\">Fjv7$F\\z$\"1%pE 5ejJ5#Fjv7$F_z$\"1fk+q`_(G#Fjv7$Fbz$\"1MMZSl1&\\#Fjv7$Fez$\"19V,yS=?FF jv7$Fhz$\"1UJFSW1CHFjv7$F[[l$\"1Y\"G*oc`_JFjv7$F^[l$\"1ixycLf(R$Fjv7$F a[l$\"1=x$yl4ik$Fjv7$Fd[l$\"1N]nCR6&*QFjv7$Fg[l$\"14lGWq5\"=%Fjv7$Fj[l $\"1]^l#pDnW%Fjv7$F]\\l$\"19DKl\"f$RZFjv7$F`\\l$\"1'e?k^\"e7]Fjv7$Fc\\ l$\"1DXTal/?`Fjv7$Ff\\l$\"10kl*Q.xh&Fjv7$Fi\\l$\"1@]o#3,v$fFjv7$F\\]l$ \"1H&=$\\*>(eiFjv7$F_]l$\"1F*Q=gsSg'Fjv7$Fb]l$\"1+/!o`ba%pFjv7$Fe]l$\" 1*\\AV(eY.tFjv7$Fh]l$\"1M.zaoRnwFjv7$F[^l$\"1\"G%=/Dh4!)Fjv7$F^^l$\"1* z+;i=5T)Fjv7$Fa^l$\"1l4YD`Ny()Fjv7$Fd^l$\"1K*\\ckg'y\"*Fjv7$Fg^l$\"1/$ pI0j,d*FjvFi^lF\\_l-F$6$7SFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFh uFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhu FhuFhuFhuFhuFhuFhuFhuFhuFhuFhuFhuF\\_l-F$6$7SFi^lFi^lFi^lFi^lFi^lFi^lF i^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^ lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lF i^lFi^lFi^lFi^lFi^lFi^lFi^lFi^lF\\_l-%+AXESLABELSG6$%!GF^il-%%VIEWG6$% (DEFAULTGFbil" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Cur ve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 2 0" "Curve 21" "Curve 22" "Curve 23" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "If the functi on f(x,y) were less simple, for example if " }{XPPEDIT 18 0 "f(x,y) \+ = x*sin(x+y);" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"-%$sinG6#,&F'F*F(F*F*" }{TEXT -1 81 ", we could compute the integral of f over the region R numerically as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(Int(Int(x*sin(x+y),y = x^2 .. x),x = 0 .. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G/8+o!#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 223 "Observe that the Maple command for integrals has the in tegration range after the function, so that the \"outer\" integration \+ range occurs last instead of first as it does when we write the double integral using integral signs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 368 "In this worksheet, our f ocus will be on setting up double integrals, not evaluating them. To \+ help you gain skill in doing this we have devised some new Maple comm ands, programmed in the subsection above, that will draw the region co rresponding to the integration ranges you specify. For example, the \+ region R appearing above was actually drawn by the new command " } {TEXT 267 25 "dydx(x=0..1, y=x^2..x):, " }{TEXT 309 43 "although we ha ven't displayed the command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 311 26 "Notice that in the comm and" }{TEXT -1 1 " " }{TEXT 310 23 "dydx(x=0..1, y=x^2..x):" }{TEXT -1 1 " " }{TEXT 312 168 "the range for the outer integral (the \"dx\" \+ integral in this case) is given first, as it is when writing integral s, although it appears last in the Maple command above." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Nat urally, we programmed a similar command " }{TEXT 313 17 "dxdy(y=...,x =...)" }{TEXT -1 142 " for double integrals where the outer integral i s with respect to y. If we reverse the order of integration in the in tegral above, we obtain " }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }{XPPEDIT 18 0 "Int(Int(f(x,y),y = x^2 .. x),x = 0 .. 1) = Int(Int(f(x,y),x = y .. sqrt(y)),y = 0 .. 1);" "6#/-%$IntG6 $-F%6$-%\"fG6$%\"xG%\"yG/F-;*$F,\"\"#F,/F,;\"\"!\"\"\"-F%6$-F%6$-F*6$F ,F-/F,;F--%%sqrtG6#F-/F-;F4F5" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Hence, we expect the command " }{TEXT 268 26 "dxdy(y=0..1, x=y ..sqrt(y))" }{TEXT -1 67 " to produce the same region R. It does, ind eed, as shown below. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "When drawing regions Maple m ay choose a viewing window that isn't exactly what we want. Another \+ new command, " }{TEXT 258 9 "setwindow" }{TEXT 259 82 ", will allow u s to specify maximum and minimum values for x and y when using the " } {TEXT 260 4 "dxdy" }{TEXT -1 6 " and " }{TEXT 261 4 "dydx" }{TEXT -1 93 " commands. For example, if you want the viewing window be x=-5..5 and y=0..4, you can enter " }{TEXT 262 20 "setwindow(-5,5,0,4):" } {TEXT -1 12 " before the " }{TEXT 263 5 "dxdy " }{TEXT -1 3 "or " } {TEXT 264 4 "dydx" }{TEXT -1 37 " command, using the colon as shown. \+ " }{TEXT 265 82 "To return to letting Maple decide on the viewing win dow, just enter the command " }{TEXT 266 14 "setwindow( ): " }{TEXT 339 39 "using the empty parentheses as shown. " }{TEXT 340 1 " " } {TEXT 279 69 "You should reset the window this way for each new exampl e or problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 23 "Let's actually use the " }{TEXT 303 4 "dxdy" }{TEXT 304 137 " command to see that we get the same region R that we discussed a bove. We will specify a view window with the setwindow command as wel l." }{TEXT 269 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "setwindow(-.5,1,-.5,1.5): dxdy(y=0..1, x= y..sqrt(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-%\"FG6$ %\"xG%\"yG/F+;F,*$-%%sqrtG6#F,\"\"\"/F,;\"\"!F3" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6:-%'CURVESG6$7S7$$!1+++++++]!#;$\" %+5\"\"!7$$!1++](oUIn%F*F+7$$!1+]7y)e&)Q%F*F+7$$!1++D\"3F'oSF*F+7$$!1+ +voXdYPF*F+7$$!1+]i:F0EMF*F+7$$!1+]7Gz))GJF*F+7$$!1+]7.5>@GF*F+7$$!1+] 7./(H]#F*F+7$$!1+]iS.x&=#F*F+7$$!1++](3\"\\f=F*F+7$$!1++vVT5s:F*F+7$$! 1++]7Xd[7F*F+7$$!1,++vV;P#*!#0)HFUF+7$$\"1***\\(=-p6jFUF+7$$\"1'**** *\\2Mg#*FUF+7$$\"1+](=xZ&\\7F*F+7$$\"1+]i:$4wb\"F*F+7$$\"1++v=#R!z=F*F +7$$\"1+]P4A@u@F*F+7$$\"1++Dchf#\\#F*F+7$$\"1+](of2L#GF*F+7$$\"1+]7yG> 6JF*F+7$$\"1++voo6AMF*F+7$$\"1*****\\xJLu$F*F+7$$\"1++v$*yddSF*F+7$$\" 1+](=cF*F+7$$\"1,+](Q(zSfF*F+7$$\"1**\\(=-,FC'F*F+7$$\"1 +]P4tFelF*F+7$$\"1****\\73\"o'oF*F+7$$\"1**\\(oz;)*=(F*F+7$$\"1,++]*44 ](F*F+7$$\"1++DJw/>yF*F+7$$\"1,](=(4bM\")F*F+7$$\"1,++vdYC%)F*F+7$$\"1 ++Dc3uc()F*F+7$$\"1+++];$R0*F*F+7$$\"1+](=-*zq$*F*F+7$$\"1+]7G:3u'*F*F +7$$\"\"\"F-F+-%'COLOURG6&%$RGBG$\"#5!\"\"F-F--F$6#7$7$$\"+++++]!#6Feu 7$$\"+yz1OA!#5Feu-F$6#7$7$$\"+++++5F[vF`v7$$\"+gwFiJF[vF`v-F$6#7$7$$\" +++++:F[vFiv7$$\"+ZL)H(QF[vFiv-F$6#7$7$$\"+++++?F[vFbw7$$\"+cf8sWF[vFb w-F$6#7$7$$\"+++++DF[vF[x7$$FfuF[vF[x-F$6#7$7$$\"+++++IF[vFcx7$$\"+wbA xaF[vFcx-F$6#7$7$$\"+++++NF[vF\\y7$$\"+&yzg\"fF[vF\\y-F$6#7$7$$\"+++++ SF[vFey7$$\"+?`bCjF[vFey-F$6#7$7$$\"+++++XF[vF^z7$$\"+NR?3nF[vF^z-F$6# 7$7$F^xF^x7$$\"+5y1rqF[vF^x-F$6#7$7$$\"+++++bF[vF^[l7$$\"+&[)>;uF[vF^[ l-F$6#7$7$$\"+++++gF[vFg[l7$$\"+'pmfu(F[vFg[l-F$6#7$7$$\"+++++lF[vF`\\ l7$$\"+XxDi!)F[vF`\\l-F$6#7$7$$\"+++++qF[vFi\\l7$$\"+l-gm$)F[vFi\\l-F$ 6#7$7$$\"+++++vF[vFb]l7$$\"+SSDg')F[vFb]l-F$6#7$7$$\"+++++!)F[vF[^l7$$ \"+7>FW*)F[vF[^l-F$6#7$7$$\"+++++&)F[vFd^l7$$\"+gWa>#*F[vFd^l-F$6#7$7$ $\"+++++!*F[vF]_l7$$\"+!)H$o[*F[vF]_l-F$6#7$7$$\"+++++&*F[vFf_l7$$\"+X VzY(*F[vFf_l-F$6$7S7$F-F-7$$\"1nmm;arz@FUF``l7$$\"1LL$e9ui2%FUFc`l7$$ \"1nmm\"z_\"4iFUFf`l7$$\"1mmmT&phN)FUFi`l7$$\"1LLe*=)H\\5F*F\\al7$$\"1 nm\"z/3uC\"F*F_al7$$\"1++DJ$RDX\"F*Fbal7$$\"1nm\"zR'ok;F*Feal7$$\"1++D 1J:w=F*Fhal7$$\"1LLL3En$4#F*F[bl7$$\"1nm;/RE&G#F*F^bl7$$\"1+++D.&4]#F* Fabl7$$\"1+++vB_Y2aF*F[el7$$\"1mm;zXu9cF*F^el7$$\"1+++]y))GeF*Fael7$$\" 1****\\i_QQgF*Fdel7$$\"1***\\7y%3TiF*Fgel7$$\"1****\\P![hY'F*Fjel7$$\" 1LLL$Qx$omF*F]fl7$$\"1+++v.I%)oF*F`fl7$$\"1mm\"zpe*zqF*Fcfl7$$\"1+++D \\'QH(F*Fffl7$$\"1KLe9S8&\\(F*Fifl7$$\"1***\\i?=bq(F*F\\gl7$$\"1LLL3s? 6zF*F_gl7$$\"1++DJXaE\")F*Fbgl7$$\"1nmmm*RRL)F*Fegl7$$\"1mm;a<.Y&)F*Fh gl7$$\"1LLe9tOc()F*F[hl7$$\"1+++]Qk\\*)F*F^hl7$$\"1LL$3dg6<*F*Fahl7$$ \"1mmmmxGp$*F*Fdhl7$$\"1++D\"oK0e*F*Fghl7$$\"1++v=5s#y*F*Fjhl7$FhtFhtF jt-F$6$7WF^`l7$$\"11U+a'H>Q(FU$\"1mmmT&)G\\aFin7$$\"1)o\"o!\\y*=?F*Fc`l7$$\"1!y!emr\"=\\#F *Ff`l7$$\"18\"4s)Rq!*GF*Fi`l7$$\"1vr?^sGRKF*F\\al7$$\"1'\\s*Hk'=`$F*F_ al7$$\"1q.o*R>7\"QF*Fbal7$$\"1)y&zfo0!3%F*Feal7$$\"1s\"zmIe9L%F*Fhal7$ $\"1:/[OnmvXF*F[bl7$$\"1Dc$QRV/y%F*F^bl7$$\"1+aqM-&4+&F*Fabl7$$\"1CSYo g)H@&F*Fdbl7$$\"15*y'GOY4aF*Fgbl7$$\"18+)Qk&*=e&F*Fjbl7$$\"130=&*HC!y& F*F]cl7$$\"14;y?B4VfF*F`cl7$$\"1()>&pU\"3FhF*Fccl7$$\"1^J*\\'4X&G'F*Ff cl7$$\"1b\\&)4(QZX'F*Ficl7$$\"16g.XJ\">h'F*F\\dl7$$\"1T*pc[??x'F*F_dl7 $$\"1O>IW\\y:pF*Fbdl7$$\"1`0mWndnqF*Fedl7$$\"1K1&[Xo=A(F*Fhdl7$$\"1Q&> #oYa`tF*F[el7$$\"1!HxO$*fJ\\(F*F^el7$$\"1MGg^`rMwF*Fael7$$\"163;qZqqxF *Fdel7$$\"1an$)eO0+zF*Fgel7$$\"1f\\oYiBT!)F*Fjel7$$\"1.E\"yL8g;)F*F]fl 7$$\"1h))f<%orH)F*F`fl7$$\"1()*zU/\\UT)F*Fcfl7$$\"1+Ee4FTS&)F*Fffl7$$ \"1ul-/UWd')F*Fifl7$$\"1-839\"3\"y()F*F\\gl7$$\"1n%)=In\\%*))F*F_gl7$$ \"1*)\\t**[t9!*F*Fbgl7$$\"1ooY_:/H\"*F*Fegl7$$\"1cW#Q(\\ZW#*F*Fhgl7$$ \"1(\\&z?jad$*F*F[hl7$$\"1Ed@JdDg%*F*F^hl7$$\"1q*4\\l!)y*F*Fghl7$$\"1o6\"H)Qw!*)*F*FjhlF\\ilFjt -%+AXESLABELSG6$Q\"x6\"Q\"yFecm-%%VIEWG6$;$!\"&F`uFht;F[dm$\"#:F`u" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "" "" "" "" "" " " "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "As y ou see, the " }{TEXT 305 4 "dydx" }{TEXT -1 511 " command will displa y the double integral in usual printed form and then draw the region. \+ It will also try to catch some common errors and tell you about them, but it isn't foolproof. If you make some unanticipated error you wil l get one of Maple's usual error messages, or some unexpected result. \+ You should also note that if you switch the upper and lower limits o f integration on one of the integral signs, the region will remain the same. However, making this switch changes the sign of the integral. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Here 's the game. We give you a region, and you have to use the " }{TEXT 306 4 "dxdy" }{TEXT -1 4 " or " }{TEXT 307 4 "dydx" }{TEXT -1 375 " co mmands to specify the limits of integration in a double integral that \+ will integrate a function F(x,y) over the given region. For some regi ons, you will have to express the integral over the region as the sum \+ of double integrals over simpler regions. We have reduced the size o f graphic displays in the worksheet to save space, so expect those yo u generate using the " }{TEXT 314 4 "dydx" }{TEXT -1 34 " and other c ommands to be larger. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 10 "Example 1." }{TEXT -1 111 " Find integration lim its for a double integral to integrate a function F(x,y) over the reg ion shown below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 234 198 198 {PLOTDATA 2 "6;-%'CURVESG6#7$7$$!+W*p,4$!#5$\"+++++5F*7$$F,!#6F+-F$6#7 $7$$!+CD&y(eF*$\"+++++?F*7$$\"+++++SF/F6-F$6#7$7$$!+W*p,4)F*$\"+++++IF *7$$\"+++++!*F/FA-F$6#7$7$$!+l^c5&*F*$F:F*7$$\"+++++;F*FL-F$6#7$7$$!\" \"\"\"!$\"+++++]F*7$$\"+++++DF*FW-F$6#7$7$$!+j^c5&*F*$\"+++++gF*7$$\"+ ++++OF*F\\o-F$6#7$7$$!+T*p,4)F*$\"+++++qF*7$$\"+++++\\F*Fgo-F$6#7$7$$! +AD&y(eF*$\"+++++!)F*7$$\"+++++kF*Fbp-F$6#7$7$$!+O*p,4$F*$FEF*7$$\"+++ ++\")F*F]q-F$6#7$7$$\"+:w1-T!#>$\"\"\"FV7$FhqFhq-F$6#7$7$$\"+W*p,4$F*$ \"+++++6!\"*7$$\"++++57FcrFar-F$6#7$7$$\"+HD&y(eF*$\"+++++7Fcr7$$\"+++ +S9FcrF]s-F$6#7$7$$\"+Y*p,4)F*$\"+++++8Fcr7$$\"++++!p\"FcrFhs-F$6#7$7$ $\"+m^c5&*F*$\"+++++9Fcr7$$\"++++g>FcrFct-F$6#7$7$Fhq$\"+++++:Fcr7$$\" ++++]AFcrF\\u-F$6#7$7$$\"+i^c5&*F*$FOFcr7$$\"++++gDFcrFgu-F$6#7$7$$\"+ Q*p,4)F*$\"+++++D&y(eF*$\"+++++=Fc r7$$\"++++SKFcrF\\w-F$6#7$7$$\"+L*p,4$F*$\"+++++>Fcr7$$\"++++5OFcrFgw- F$6$7en7$FVFV7$$!170Q#=y_O\"!#;$\"1LLLL3VfV!#<7$$!159Ff*)GLDFcx$\"1nmm \"H[D:)Ffx7$$!1/_QC.6.QFcx$\"1LLLe0$=C\"Fcx7$$!1,.')Q4U7]Fcx$\"1LLL3RB r;Fcx7$$!13DMpceDhFcx$\"1mm;zjf)4#Fcx7$$!1Qv;+FafqFcx$\"1LL$e4;[\\#Fcx 7$$!1FBz[!>8\"zFcx$\"1++]i'y]!HFcx7$$!1b[3Qh-a')Fcx$\"1LL$ezs$HLFcx7$$ !1E?Z5bcT#*Fcx$\"1++]7iI_PFcx7$$!10w/M\"oen*Fcx$\"1nmm;_M(=%Fcx7$$!1Z8 [pzD5)*Fcx$\"1++]7l$*yVFcx7$$!1/9\\jt64**Fcx$\"1LLL3y_qXFcx7$$!1tzN\"G au(**Fcx$\"1nm;HU@'y%Fcx7$$!0N+&Fcx7$$!1/R!fFb k(**Fcx$\"1++++FZ=_Fcx7$$!1MK#zGZn!**Fcx$\"1+++]Z/NaFcx7$$!1H(>Kl*='z* Fcx$\"1******\\?vVcFcx7$$!1'><)\\D`V'*Fcx$\"1+++]$fC&eFcx7$$!1msyOl'3E *Fcx$\"1LL$ez6:B'Fcx7$$!1#zgW2&oN')Fcx$\"1mmm;=C#o'Fcx7$$!1W#[,/I-(zFc x$\"1mmmm#pS1(Fcx7$$!1<5bi?x_qFcx$\"1++]i`A3vFcx7$$!1AO(oQZc7'Fcx$\"1m mmm(y8!zFcx7$$!1rJX7/k,]Fcx$\"1++]i.tK$)Fcx7$$!1p%fK#HvXQFcx$\"1++](3z Mu)Fcx7$$!1y0DIQ%=d#Fcx$\"1nmm\"H_?<*Fcx7$$!1Y%R*3]Ug8Fcx$\"1nm;zihl&* Fcx7$$!1CAHY)485$!#=$\"1LLL3#G,***Fcx7$$\"1l$f>&=8]8Fcx$\"1LLezw5V5Ff \\l7$$\"1qP)R-$GKDFcx$\"1++v$Q#\\\"3\"Ff\\l7$$\"1g90KNAnPFcx$\"1LL$e\" *[H7\"Ff\\l7$$\"1XaI#)3zv\\Fcx$\"1+++qvxl6Ff\\l7$$\"1zK4lUCrgFcx$\"1++ ]_qn27Ff\\l7$$\"19uU3yMJqFcx$\"1++Dcp@[7Ff\\l7$$\"1.*)f%eGL'zFcx$\"1++ ]2'HKH\"Ff\\l7$$\"1NJ))3Mil')Fcx$\"1nmmwanL8Ff\\l7$$\"1[W+Q***4E*Fcx$ \"1+++v+'oP\"Ff\\l7$$\"1,nX%p[Pl*Fcx$\"1LLeR<*fT\"Ff\\l7$$\"1whUR682)* Fcx$\"1n;HiBQP9Ff\\l7$$\"1?bH#*>C;**Fcx$\"1+++&)Hxe9Ff\\l7$$\"1Nx7!fP! 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Then see if you c an describe the boundaries algebraically. Use the " }{TEXT 316 4 "dxd y" }{TEXT -1 8 " and/or " }{TEXT 317 4 "dydx" }{TEXT -1 40 " commands \+ to try out your answers. Use " }{TEXT 318 13 "setwindow( ):" }{TEXT -1 44 " to have Maple decide on the view window. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 271 102 "You c an get hints by clicking on the hidden sections below, but experiment yourself before peeking!! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "setwindow( ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 " Hint 1" }}{PARA 0 "" 0 "" {TEXT -1 8 "Use dxdy" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "Hint 2" }{TEXT 274 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The equation of the curve on t he left has the form x = A*sin(B*y). What are A and B?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "Hint 3" }}{PARA 0 "" 0 "" {TEXT -1 38 " Th e curve on the right has equation " }{XPPEDIT 18 0 "y = sqrt(x);" "6# /%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution\n" }}{PARA 0 "" 0 "" {TEXT 281 46 "setwindow():dx dy(y=0..2,x=-sin(Pi*y)..y^2); " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "This command corresponds the the double integral " }} {PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Int(Int(f(x,y),x \+ = -sin(Pi*y) .. y^2),x = 0 .. 2);" "6#-%$IntG6$-F$6$-%\"fG6$%\"xG%\"yG /F+;,$-%$sinG6#*&%#PiG\"\"\"F,F5!\"\"*$F,\"\"#/F+;\"\"!F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 319 11 "Example 2. " }{TEXT -1 235 " To integrate over the L-shaped re gion below we might break the region into two rectangular regions and \+ add the integrals over these rectangular regions. For example, the in tegral of F(x,y) over the region shown below can be written \n" } {XPPEDIT 18 0 "Int(Int(F(x,y),x = 0 .. 1),y = 0 .. 2);" "6#-%$IntG6$-F $6$-%\"FG6$%\"xG%\"yG/F+;\"\"!\"\"\"/F,;F/\"\"#" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Int(Int(F(x,y),x = 0 .. 3),y = 2 .. 4);" "6#-%$IntG6$-F $6$-%\"FG6$%\"xG%\"yG/F+;\"\"!\"\"$/F,;\"\"#\"\"%" }{TEXT -1 43 " and \+ we can draw the region by the command\n" }{TEXT 282 52 "display(dxdy(y =0..2,x=0..1), dxdy(y=2..4,x=0..3));\n\n" }{TEXT 283 24 "Note how the \+ we use the " }{TEXT 284 7 "display" }{TEXT 285 9 " command " }{TEXT -1 35 "to combine the results of the two " }{TEXT 286 4 "dxdy" } {TEXT -1 12 " commands. \n" }}{PARA 0 "" 0 "" {TEXT -1 65 "How can you write the integral over the region as the sum of two " }{TEXT 320 6 " \"dydx\"" }{TEXT -1 14 " (rather than " }{TEXT 321 7 "\"dxdy\")" } {TEXT -1 42 " double integrals? How could you use the " }{TEXT 326 7 "display" }{TEXT -1 18 " command with two " }{TEXT 287 4 "dydx" } {TEXT -1 32 " commands to draw the region?\n\n\n" }{GLPLOT2D 259 233 233 {PLOTDATA 2 "6R-%'CURVESG6#7$7$\"\"!$\"+++++5!#57$$\"\"\"F(F)-F$6# 7$7$F($\"+++++?F+7$F-F3-F$6#7$7$F($\"+++++IF+7$F-F:-F$6#7$7$F($\"+++++ SF+7$F-FA-F$6#7$7$F($\"+++++]F+7$F-FH-F$6#7$7$F($\"+++++gF+7$F-FO-F$6# 7$7$F($\"+++++qF+7$F-FV-F$6#7$7$F($\"+++++!)F+7$F-Fgn-F$6#7$7$F($\"+++ ++!*F+7$F-F^o-F$6#7$7$F(F-7$F-F--F$6#7$7$F($\"+++++6!\"*7$F-Fjo-F$6#7$ 7$F($\"+++++7F\\p7$F-Fbp-F$6#7$7$F($\"+++++8F\\p7$F-Fip-F$6#7$7$F($\"+ ++++9F\\p7$F-F`q-F$6#7$7$F($\"+++++:F\\p7$F-Fgq-F$6#7$7$F($\"+++++;F\\ p7$F-F^r-F$6#7$7$F($\"+++++F\\p7$F-Fcs-F$6$7S7$F(F(7$F($\"1LLLL3VfV!#<7$F($ \"1nmm\"H[D:)F]t7$F($\"1LLLe0$=C\"!#;7$F($\"1LLL3RBr;Fdt7$F($\"1mm;zjf )4#Fdt7$F($\"1LL$e4;[\\#Fdt7$F($\"1++]i'y]!HFdt7$F($\"1LL$ezs$HLFdt7$F ($\"1++]7iI_PFdt7$F($\"1nmm;_M(=%Fdt7$F($\"1LLL3y_qXFdt7$F($\"1+++]1!> +&Fdt7$F($\"1+++]Z/NaFdt7$F($\"1+++]$fC&eFdt7$F($\"1LL$ez6:B'Fdt7$F($ \"1mmm;=C#o'Fdt7$F($\"1mmmm#pS1(Fdt7$F($\"1++]i`A3vFdt7$F($\"1mmmm(y8! 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Use the " }{TEXT 273 4 " dxdy" }{TEXT -1 5 " and " }{TEXT 275 4 "dydx" }{TEXT -1 1 " " }{TEXT 276 8 "commands" }{TEXT -1 63 " to experiment, check your work and pro duce your final answer. " }{TEXT 277 129 "Remember that if you use the setwindow command, the view window will remain in effect until you re lease it by using the command: " }{TEXT 278 15 "setwindow( ):. " } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 10 "Problem 1." }{TEXT -1 14 " Give both \"" }{TEXT 325 5 "dxdy\"" }{TEXT -1 50 " and \"dydx\" integrals of F(x,y) over the reg ion: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 2 ". " }{GLPLOT2D 244 188 188 {PLOTDATA 2 "6;-%'CURVESG6#7$7$$\"+++++5!#5F(7$$\"+++++?F*F(-F$6#7$7$F,F,7$$\"++ +++SF*F,-F$6#7$7$$\"+++++IF*F97$$\"+++++gF*F9-F$6#7$7$F3F37$$\"+++++!) 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" 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 327 11 "Problem 2. " }{TEXT -1 103 " The curved boundary is a parabola. 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" }{TEXT -1 78 " The curved part is an arc of a circle. Give both \"dxdy\" and \"d ydx\" integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{GLPLOT2D 203 198 198 {PLOTDATA 2 "6;-%'CURVESG6#7$7$$\"+++++5!#5$!+O%)\\(*>!\"*7$F($!++ +++>F--F$6#7$7$$\"+++++?F*$!+u[(**)>F-7$F5$!+++++=F--F$6#7$7$$\"+++++I F*$!+$*>Px>F-7$F@$!+++++F-7$FK$!+++++ ;F--F$6#7$7$$\"+++++]F*$!+t;\\O>F-7$FV$!+++++:F--F$6#7$7$$\"+++++gF*$! +.%yy!>F-7$F[o$!+++++9F--F$6#7$7$$\"+++++qF*$!++%*\\t=F-7$Ffo$!+++++8F --F$6#7$7$$\"+++++!)F*$!+y-.L=F-7$Fap$!+++++7F--F$6#7$7$$\"+++++!*F*$! +5r0'y\"F-7$F\\q$!+++++6F--F$6#7$7$$\"\"\"\"\"!$!+330K:F-7$Fgs$!+++++qF*-F$6#7$ 7$$\"+++++9F-$!+'o&GG9F-7$Fbt$!+++++gF*-F$6#7$7$$\"+++++:F-$!+cc(GK\"F -7$F]u$!+++++]F*-F$6#7$7$$\"+++++;F-Ffp7$Fhu$!+++++SF*-F$6#7$7$$\"++++ +F-$!+)*z*\\C'F*7$Fgw$!+++++5F*-F$6$7Y7$Fiq$! \"#Fiq7$$\"1LLLL3VfV!#<$!1QIiF[_**>!#:7$$\"1nmm\"H[D:)Fgx$!1:_53xL)*>F jx7$$\"1LLLe0$=C\"!#;$!1*\\M!>49'*>Fjx7$$\"1LLL3RBr;Fcy$!1C=H6_+$*>Fjx 7$$\"1mm;zjf)4#Fcy$!152ge#f*))>Fjx7$$\"1LL$e4;[\\#Fcy$!1$>mbsyV)>Fjx7$ $\"1++]i'y]!HFcy$!1&eLj\"))yy>Fjx7$$\"1LL$ezs$HLFcy$!1\"4j()\\$4s>Fjx7 $$\"1++]7iI_PFcy$!1F8Up^[k>Fjx7$$\"1nmm;_M(=%Fcy$!1Tk&\\:uc&>Fjx7$$\"1 LLL3y_qXFcy$!1m%GlVvq%>Fjx7$$\"1+++]1!>+&Fcy$!1%H#e)eUk$>Fjx7$$\"1+++] Z/NaFcy$!1I]W(\\MZ#>Fjx7$$\"1+++]$fC&eFcy$!1V;D-eX7>Fjx7$$\"1LL$ez6:B' Fcy$!1%)H$)RFW+>Fjx7$$\"1mmm;=C#o'Fcy$!14ce%pm])=Fjx7$$\"1mmmm#pS1(Fcy $!17)[R1$4r=Fjx7$$\"1++]i`A3vFcy$!1&4Frn;P&=Fjx7$$\"1mmmm(y8!zFcy$!16C d^HIP=Fjx7$$\"1++]i.tK$)Fcy$!1`bo@j9==Fjx7$$\"1++](3zMu)Fcy$!1#=)QRRv) z\"Fjx7$$\"1nmm\"H_?<*Fcy$!10*[r\"HGxceSszFcy7$$\"1LLL`v&Q(=Fjx$!1\"ymAy1/*pFcy7$$\"1++ DOl5;>Fjx$!1O[CXP!=t&Fcy7$$\"1+++q`KO>Fjx$!1%GE@ZOk+&Fcy7$$\"1++v.Uac> Fjx$!1]tQ%)=]YTFcy7$$\"1+D\"G:3u'>Fjx$!1n)[Fjx $!1e1=p$f+%HFcy7$$\"1]iSwSq$)>Fjx$!1&Qe.z.za#Fcy7$$\"1+v$40O\"*)>Fjx$! 1u))p#es<3#Fcy7$$\"1](oa-oX*>Fjx$!1PAj)RPIZ\"Fcy7$$\"\"#FiqFiq-%'COLOU RG6&%$RGBG$\"#5F^rFiqFiq-F$6$7SFax7$Fex$!1nmm\"p0k&>Fjx7$F\\y$!1LL3Fjx7$Fay$!1nm;Wp\"e(=Fjx7$Fgy$!1nm;4m(G$=Fjx7$F\\z$!1LL3i.9!z\"Fjx7$ Faz$!1nmT!R=0v\"Fjx7$Ffz$!1++vL@\\4s%Ha\"Fjx7$F_\\l$!1+++N *4)*\\\"Fjx7$Fd\\l$!1+++Db\\c9Fjx7$Fi\\l$!1+++lSv99Fjx7$F^]l$!1nmT?)[o P\"Fjx7$Fc]l$!1LLL=exJ8Fjx7$Fh]l$!1LLLtIf$H\"Fjx7$F]^l$!1++vju<\\7Fjx7 $Fb^l$!1MLLB@')47Fjx7$Fg^l$!1++vjpsm6Fjx7$F\\_l$!1++D\"4_c7\"Fjx7$Fa_l $!1LL$3x%z#3\"Fjx7$Ff_l$!1LL3s$QM/\"Fjx7$F[`l$!1nm;zr)4+\"Fjx7$F``l$!1 om;/K#*o&*Fcy7$Fe`l$!1,+]ih2&=*Fcy7$Fj`l$!1ommT3^q()Fcy7$F_al$!1,+++VA U$)Fcy7$Fdal$!1-++v%HK#zFcy7$Fial$!1-+]P/$y^(Fcy7$F^bl$!1-++DRqnqFcy7$ Fcbl$!1NLLL_CjmFcy7$Fhbl$!1+++]#*RJiFcy7$F]cl$!1om;/E3SeFcy7$Fbcl$!1++ +],F7aFcy7$Fgcl$!1NL$3(>t4]Fcy7$F\\dl$!1,+](ej*)e%Fcy7$Fadl$!1MLL$e&ex TFcy7$Ffdl$!1,+]P4\"pu$Fcy7$F[el$!1mmmm+7KLFcy7$F`el$!1nmm\"\\Oz!HFcy7 $Feel$!1ML$3Pls[#Fcy7$Fjel$!1,+++Br+@Fcy7$F_fl$!1MLLe)ywl\"Fcy7$Fdfl$! 1nmmmWUh7Fcy7$Fifl$!1&****\\PY$*Q)Fgx7$Fcgl$!1'****\\izbM%FgxF`ilFcil- F$6$7SFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxF axFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFaxFa xFaxFaxFaxFaxFcil-F$6$7SF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF` ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`il F`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF`ilF` ilF`ilF`ilFcil-%+AXESLABELSG6$%!GFbcm-%%VIEWG6$%(DEFAULTGFfcm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "" "" "" "" "" "" " " "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 329 12 "Pro blem 4. " }{TEXT -1 39 "Give both \"dxdy\" and \"dydx\" integrals.\n " }{GLPLOT2D 259 233 233 {PLOTDATA 2 "6R-%'CURVESG6#7$7$$!+++++Q!\"*\" \"!7$F($\"+++++5!#5-F$6#7$7$$!+++++OF*F+7$F4$\"+++++?F/-F$6#7$7$$!++++ +MF*F+7$F=$\"+++++IF/-F$6#7$7$$!+++++KF*F+7$FF$\"+++++SF/-F$6#7$7$$!\" $F+F+7$FO$\"+++++]F/-F$6#7$7$$!+++++GF*F+7$FX$\"+++++gF/-F$6#7$7$$!+++ ++EF*F+7$F[o$\"+++++qF/-F$6#7$7$$!+++++CF*F+7$Fdo$\"+++++!)F/-F$6#7$7$ $!+++++AF*F+7$F]p$\"+++++!*F/-F$6#7$7$$!\"#F+F+7$Ffp$\"\"\"F+-F$6#7$7$ $!+++++=F*F+7$F_q$\"+++++6F*-F$6#7$7$$!+++++;F*F+7$Fhq$\"+++++7F*-F$6# 7$7$$!+++++9F*F+7$Far$\"+++++8F*-F$6#7$7$$!+++++7F*F+7$Fjr$\"+++++9F*- F$6#7$7$$!\"\"F+F+7$Fcs$\"+++++:F*-F$6#7$7$$!+++++!)F/F+7$F\\t$\"+++++ ;F*-F$6#7$7$$!+++++gF/F+7$Fet$\"+++++F*-F$6$7S7$$!\"%F+F+7$$!1MLL $Q6G\"R!#:F+7$$!1nm;M!\\p$QFevF+7$$!1LLL))Qj^PFevF+7$$!1LLL=KvlOFevF+7 $$!1nm;C2G!e$FevF+7$$!1LL$3yO5]$FevF+7$$!1++]nU)*=MFevF+7$$!1ML$3WDTL$ FevF+7$$!1++]d(Q&\\KFevF+7$$!1nmmc4`iJFevF+7$$!1LLLQW*e3$FevF+7$$!1+++ q)>'**HFevF+7$$!1+++]5*H\"HFevF+7$$!1+++I\"3&HGFevF+7$$!1LL$3k(p`FFevF +7$$!1nmmO;bjEFevF+7$$!1nmmYh=(e#FevF+7$$!1++]F\\N)\\#FevF+7$$!1nmmYUs >CFevF+7$$!1++]FRXLBFevF+7$$!1++]#=/8D#FevF+7$$!1nmmT&*el@FevF+7$$!1nm ;Wn(o3#FevF+7$$!1LLLeV(>+#FevF+7$$!1ML$3k%y8>FevF+7$$!1++]K_,P=FevF+7$ $!1MLLo@5a+\"FevF+7$$!1-++vr#z<*! #;F+7$$!1ommm6+&Fb]l7$Fhx$\"1+++]Z/NaFb]l7$F[y$\"1,++]$fC &eFb]l7$F^y$\"1LL$ez6:B'Fb]l7$Fay$\"1lmm;=C#o'Fb]l7$Fdy$\"1mmmm#pS1(Fb ]l7$Fgy$\"1)***\\i`A3vFb]l7$Fjy$\"1lmmm(y8!zFb]l7$F]z$\"1++]i.tK$)Fb]l 7$F`z$\"1****\\(3zMu)Fb]l7$Fcz$\"1mmm\"H_?<*Fb]l7$Ffz$\"1nm;zihl&*Fb]l 7$Fiz$\"1LLL3#G,***Fb]l7$F\\[l$\"1LLezw5V5Fev7$F_[l$\"1++v$Q#\\\"3\"Fe v7$Fb[l$\"1LL$e\"*[H7\"Fev7$Fe[l$\"1+++qvxl6Fev7$Fh[l$\"1++]_qn27Fev7$ F[\\l$\"1++Dcp@[7Fev7$F^\\l$\"1++]2'HKH\"Fev7$Fa\\l$\"1nmmwanL8Fev7$Fd \\l$\"1+++v+'oP\"Fev7$Fg\\l$\"1LLeR<*fT\"Fev7$Fj\\l$\"1+++&)Hxe9Fev7$F ]]l$\"1mm\"H!o-*\\\"Fev7$F`]l$\"1++DTO5T:Fev7$Fd]l$\"1nmmT9C#e\"Fev7$F g]l$\"1++D1*3`i\"Fev7$Fj]l$\"1LLL$*zym;Fev7$F]^l$\"1LL$3N1#4Fev7$F__l$\"1++v.Uac>Fev7$F+$\"\"#F+ Fc_l-F$6$7SF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_ vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_vF_v F_vF_vF_vF_vF_vF_vFc_l-F$6$7SFb_l7$F+$\"1LLLL3VfVFa_l7$F+$\"1nmm\"H[D: )Fa_l7$F+$\"1LLLe0$=C\"Fb]l7$F+$\"1LLL3RBr;Fb]l7$F+Fi`l7$F+$\"1LL$e4;[ \\#Fb]l7$F+F_al7$F+$\"1LL$ezs$HLFb]l7$F+$\"1++]7iI_PFb]l7$F+Fhal7$F+F[ bl7$F+F^bl7$F+Fabl7$F+$\"1+++]$fC&eFb]l7$F+Fgbl7$F+$\"1mmm;=C#o'Fb]l7$ F+F]cl7$F+$\"1++]i`A3vFb]l7$F+$\"1mmmm(y8!zFb]l7$F+Ffcl7$F+$\"1++](3zM u)Fb]l7$F+$\"1nmm\"H_?<*Fb]l7$F+F_dl7$F+Fbdl7$F+Fedl7$F+Fhdl7$F+F[el7$ F+F^el7$F+Fael7$F+Fdel7$F+Fgel7$F+Fjel7$F+F]fl7$F+F`fl7$F+Fcfl7$F+Fffl 7$F+Fifl7$F+F\\gl7$F+F_gl7$F+Fbgl7$F+Fegl7$F+Fhgl7$F+F[hl7$F+F^hl7$F+F ahl7$F+Fdhl7$F+FghlFihlFc_l-F$6#7$7$F@F+7$F@Fju-F$6#7$7$FenF+7$FenFau- F$6#7$7$F`pF+7$F`pFht-F$6#7$7$F[rF+7$F[rF_t-F$6#7$7$FfsF+7$FfsFfs-F$6# 7$7$FauF+7$FauF]s-F$6#7$7$$\"+++++@F*F+7$F]`mFdr-F$6#7$7$$\"+++++CF*F+ 7$Fd`mF[r-F$6#7$7$$\"+++++FF*F+7$F[amFbq-F$6#7$7$$\"\"$F+F+7$FbamFip-F $6#7$7$$\"+++++LF*F+7$FiamF`p-F$6#7$7$$\"+++++OF*F+7$F`bmFgo-F$6#7$7$$ \"+++++RF*F+7$FgbmF^o-F$6#7$7$$\"+++++UF*F+7$F^cmFen-F$6#7$7$$\"+++++X F*F+7$FecmFR-F$6#7$7$$\"+++++[F*F+7$F\\dmFI-F$6#7$7$$\"+++++^F*F+7$Fcd mF@-F$6#7$7$$\"+++++aF*F+7$FjdmF7-F$6#7$7$$\"+++++dF*F+7$FaemF--F$6$7S Fb_l7$$\"1+++]#HyI\"Fb]lF+7$$\"1++]([kdW#Fb]lF+7$$\"1+++v;\\DPFb]lF+7$ $\"1+++Dq0]\"FevF+7$$\"1+++DM^I;FevF+7$$ \"1+++0ytb@F evF+7$$\"1++v3wY_AFevF+7$$\"1+++IOTqBFevF+7$$\"1++v3\">)*\\#FevF+7$$\" 1++DEP/BEFevF+7$$\"1++](o:;v#FevF+7$$\"1++v$)[opGFevF+7$$\"1++]i%Qq*HF evF+7$$\"1++vQIKHJFevF+7$$\"1++D^rZWKFevF+7$$\"1++]Zn%)oLFevF+7$$\"1++ +5FL(\\$FevF+7$$\"1++]d6.BOFevF+7$$\"1++vo3lWPFevF+7$$\"1++]A))ozQFevF +7$$\"1+++Ik-,SFevF+7$$\"1+++D-eITFevF+7$$\"1++v=_(zC%FevF+7$$\"1+++b* =jP%FevF+7$$\"1++v3/3(\\%FevF+7$$\"1++vB4JBYFevF+7$$\"1+++DVsYZFevF+7$ $\"1++v=n#f([FevF+7$$\"1+++!)RO+]FevF+7$$\"1++]_!>w7&FevF+7$$\"1++v)Q? QD&FevF+7$$\"1+++5jyp`FevF+7$$\"1++]Ujp-bFevF+7$$\"1+++gEd@cFevF+7$$\" 1++v3'>$[dFevF+7$$\"1++D6EjpeFevF+7$$\"\"'F+F+Fc_l-F$6$7SFihl7$Fhem$\" 1nmm\"p0k&>Fev7$F[fm$\"1LL3Fev7$F^fm$\"1nm;Wp\"e(=Fev7$Fafm$\"1nm ;4m(G$=Fev7$Fdfm$\"1LL3i.9!z\"Fev7$Fgfm$\"1nmT!R=0v\"Fev7$Fjfm$\"1++vL @\\4s%Ha\"Fev7$Figm$\"1+++N*4)*\\\"Fev7$F\\hm$\"1+++Db \\c9Fev7$F_hm$\"1+++lSv99Fev7$Fbhm$\"1nmT?)[oP\"Fev7$Fehm$\"1LLL=exJ8F ev7$Fhhm$\"1LLLtIf$H\"Fev7$F[im$\"1++vju<\\7Fev7$F^im$\"1MLLB@')47Fev7 $Faim$\"1++vjpsm6Fev7$Fdim$\"1++D\"4_c7\"Fev7$Fgim$\"1LL$3x%z#3\"Fev7$ Fjim$\"1LL3s$QM/\"Fev7$F]jm$\"1nm;zr)4+\"Fev7$F`jm$\"1om;/K#*o&*Fb]l7$ Fcjm$\"1,+]ih2&=*Fb]l7$Ffjm$\"1ommT3^q()Fb]l7$Fijm$\"1,+++VAU$)Fb]l7$F \\[n$\"1-++v%HK#zFb]l7$F_[n$\"1-+]P/$y^(Fb]l7$Fb[n$\"1-++DRqnqFb]l7$Fe [n$\"1NLLL_CjmFb]l7$Fh[n$\"1+++]#*RJiFb]l7$F[\\n$\"1om;/E3SeFb]l7$F^\\ n$\"1+++],F7aFb]l7$Fa\\n$\"1NL$3(>t4]Fb]l7$Fd\\n$\"1,+](ej*)e%Fb]l7$Fg \\n$\"1MLL$e&exTFb]l7$Fj\\n$\"1,+]P4\"pu$Fb]l7$F]]n$\"1mmmm+7KLFb]l7$F `]n$\"1nmm\"\\Oz!HFb]l7$Fc]n$\"1ML$3Pls[#Fb]l7$Ff]n$\"1,+++Br+@Fb]l7$F i]n$\"1MLLe)ywl\"Fb]l7$F\\^n$\"1nmmmWUh7Fb]l7$F_^n$\"1&****\\PY$*Q)Fa_ l7$Fb^n$\"1'****\\izbM%Fa_lFd^nFc_lF_il-F$6$7SFd^nFd^nFd^nFd^nFd^nFd^n Fd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd ^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^n Fd^nFd^nFd^nFd^nFd^nFd^nFd^nFd^nFc_l-%+AXESLABELSG6$%!GF]hn-%%VIEWG6$% (DEFAULTGFahn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" " " "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" "" " " }}{TEXT -1 2 "\n\n" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 280 0 " " }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 26 "Part II, Polar Coordinates" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "In the s ubsection at the beginning of this worksheet, we have programmed two o ther commands, " }{TEXT 288 8 "drdtheta" }{TEXT -1 6 ", and " }{TEXT 289 10 "dthetadr, " }{TEXT 290 79 "that draw regions corresponding to \+ double integrals in polar coordinates. The " }{TEXT 291 9 "setwindow " }{TEXT 292 210 " command has no effect on these polar coordinate com mands. You will find it helpful to know a little more about plotting certain curves in polar coordinates, especially in example 2 below. \+ Click the + below " }{TEXT -1 27 "for more on polar plotting." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Polar plotting in brief " }}{PARA 0 "" 0 "" {TEXT -1 36 "Execute the following command first. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Curves are frequently described in polar coordinates b y equations like " }{XPPEDIT 18 0 "r = f(theta);" "6#/%\"rG-%\"fG6#%&t hetaG" }{TEXT -1 28 " with a range of values for " }{XPPEDIT 18 0 "the ta;" "6#%&thetaG" }{TEXT -1 105 ". The corresponding curve is then th e set of points in the plane that have a pair of polar coordinates (" }{TEXT 294 2 "r," }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 46 ") satisfying the equation. We agree that if " }{TEXT 295 3 "r " } {TEXT -1 26 "is negative, we identify (" }{TEXT 256 2 "r," }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 42 ") with the point we get by r eflecting (-" }{TEXT 256 2 "r," }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 127 ") through the origin, so for example, the point with re ctangular coordinates (1,1) can be represented in polar coordinates as (" }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 13 " ) or \+ as ( -" }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "5*Pi/4;" "6#*(\"\"&\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT -1 5 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The easiest example is provided by the equation " }{XPPEDIT 18 0 "r = c;" "6#/%\"rG%\"cG" }{TEXT -1 8 " where " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 51 " is a positive constant, i.e. where the functi on f(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 79 ") is constan t. The corresponding curve is then obviously a circle of radius " } {XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 23 ". We can plot r= f(" } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 50 ") by plotting the po int with polar coordinates (f(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 3 "), " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 12 ") \+ for each " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 61 " in the given range. Thus we can think of varying the angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 61 " and, for each angle, plotting t he point lying at distance f(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 45 ") from the origin along the ray making angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 33 " with the positive x-axis. I f f(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 68 ") is negative we go backward along the ray. Maple uses the command " }{TEXT 296 11 "polarplot " }{TEXT 297 42 "to do this. Study the following examp les." }}{PARA 0 "" 0 "" {TEXT 298 2 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "To plot " }{XPPEDIT 18 0 "r = 5+sin(8*theta);" "6#/%\"rG, &\"\"&\"\"\"-%$sinG6#*&\"\")F'%&thetaGF'F'" }{TEXT -1 6 " for " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 13 " from 0 to 2" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "polarplot(5+sin(8 *theta), theta=0..2*Pi); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6$7gv7$$\"\"&\"\"!F*7$$\"1c[a'H!*y]&!#:$\"18B \"HW(exP!#;7$$\"18x6tz2MeF.$\"1^rNJ!G//)F17$$\"1!)QS.^D*)eF.$\"1Mr(\\% G<7**F17$$\"1)el)3-\\%)eF.$\"1Qum-!\\;<\"F.7$$\"1X)y47)3>eF.$\"1KGh%y& )*R8F.7$$\"1%\\8#eF.7$$\"1KA6^VdSSF.$\"1ypmd#\\(*)>F .7$$\"1&*G;+2CsNF.$\"1L,EKRFp?F.7$$\"1S:tAU`([$F.$\"1!y2,C]\"*4#F.7$$ \"1f(R<x w>>LF.$\"1o%*f?L!GB#F.7$$\"1`_3J_P\"H$F.$\"1o(z&4XC&H#F.7$$\"1EzrpJzwK F.$\"1]H)*yqynBF.7$$\"1;1#z7!\\uKF.$\"1#>9.j03X#F.7$$\"1ouw;!eKG$F.$\" 1b@ewyVWDF.7$$\"1G*fMnUoQ$F.$\"1`P!*31ozHF.7$$\"1@71j(f?`$F.$\"1w40NSd ?NF.7$$\"12'\\v2S^i$F.$\"1Kl.]sx6TF.7$$\"1')efx$)zyNF.$\"1BL`Q0**GYF.7 $$\"1()=e32TQNF.$\"1QHCR+$yt%F.7$$\"1p)y:6,f[$F.$\"1A$pie_M$[F.7$$\"1U V\"zWb9U$F.$\"1DnoghB%HO!3&F.7$$\"1\"*Q@DO!=%HF.$\"1^=T!fV23&F.7$$\"126XxdmMCF.$\"1 5&G8I\"yL\\F.7$$\"1&3Riiby!>F.$\"1?EvRiU:YF.7$$\"1wU/8g>B9F.$\"1Dih$*f 'GC%F.7$$\"1!pW0>Cx.\"F.$\"1lD6_)yf(RF.7$$\"1EMlkRJj'*F1$\"1-nFi4TWRF. 7$$\"1$Gi$H)4;**)F1$\"1yu-eC&[#RF.7$$\"1Vf-F5Rd$)F1$\"1[quoJ&z\"RF.7$$ \"1#Rldj%=bxF1$\"1kZ&**>^T#RF.7$$\"1jI+Z'G*yrF1$\"1\"[!oXYmVRF.7$$\"1( HkOoF@i'F1$\"1Ne_#*o[wRF.7$$\"1]u`fd,ygF1$\"1Zv^3cQASF.7$$\"1>&ekkA(Rb F1$\"1Xk>Za!43%F.7$$\"1..Y+177IF1$\"1#>V8%31yWF.7$$!1kHGqrQ))H!#=$\"1[ HuXfx/]F.7$$!1\"odqM'*py$F1$\"1`g&o..*3bF.7$$!1V8WeCbA!)F1$\"1DY2\"=zK $eF.7$$!1&f*4)*4fd!*F1$\"12j(*ex=reF.7$$!1k\\NE]*z+\"F.$\"1_=S$eN8*eF. 7$$!13W/;L.36F.$\"1nR%='G@$*eF.7$$!1(ez+*f'\\?\"F.$\"1'QdU0Em(eF.7$$!1 m,v#=KzH\"F.$\"1t0;$3+<%eF.7$$!1lsX*pUhQ\"F.$\"1p#\\uMu))y&F.7$$!1o_!Q 8.*o9F.$\"1'[EWk#*)=dF.7$$!1)=dq&\\jX:F.$\"1*pfsA(yKcF.7$$!1Jj'Qg:#oF.$\"18W.c#\\Mn%F.7$$!1)=.(fW&)*)>F.$\" 1%)R/eXoRSF.7$$!1)4ohQ[\\2#F.$\"1p_7lak`NF.7$$!1P?YddY-@F.$\"1?$QO!e() zMF.7$$!1\"p;!)\\e`8#F.$\"1g*eJ`HF.$\"1x>8\"e4%zLF.7$$!1h`$[e7$fNF.$\"1%= ,0!\\'4a$F.7$$!1]'3zf0`7%F.$\"1f.6*3+fi$F.7$$!1)*RpY(R4i%F.$\"1dj<%[j6 e$F.7$$!1-@:?&oBt%F.$\"123'y'='3a$F.7$$!1Pu#=v4-$[F.$\"1n0b#3H!)[$F.7$ $!1wIPJd>8\\F.$\"1yeU]#\\GU$F.7$$!1(>*=%)fJ!)\\F.$\"1v$pnc_dM$F.7$$!1x O'3JV3.&F.$\"1)Qr6u)RdKF.7$$!1yM&[2aV1&F.$\"1YV=:bleJF.7$$!1[,Y:Au!3&F .$\"1>T%y%Gd]IF.7$$!1rB/)fE-3&F.$\"1o*3TX`V$HF.7$$!1r0TVi+P\\F.$\"1&=D K(R`TCF.7$$!1a;0\"*efIYF.$\"1MU!*>*y\"H>F.7$$!1+JR9z4iUF.$\"1qs!>)\\jZ 9F.7$$!12?$p9a%))RF.$\"10hy$ft91\"F.7$$!1vK$33FC&RF.$\"1@Ezm+On)*F17$$ !1,!)zEs#)GRF.$\"1h:)\\N;e;*F17$$!13fD$R?%=RF.$\"1\"H@u^>_])F17$$!17!G /!*R<#RF.$\"11#\\[dE)zyF17$$!1+[3-h2RRF.$\"1\"REJT*>$G(F17$$!1I;A'3q/( RF.$\"1$QyMR!Q3nF17$$!1?We$o6d,%F.$\"1)fH;1x![hF17$$!1\"=cd$=MuSF.$\"1 ywSfWF.$\"1XQJ2x)p6$F17$$!1aJ+/\"o^(\\F.$\"1Y!=\"* *4'Ha\"!#<7$$!1V(HXG')G\\&F.$!1LO9vcbROF17$$!1g%=cz!pHeF.$!1)fElG%eVzF 17$$!1\"p#)*G/?))eF.$!1)Q5c]6%R)*F17$$!1u2W*[La)eF.$!1yQ$*fB%o;\"F.7$$ !1k2;bpmgeF.$!1>D_h$GSD\"F.7$$!1DU#*>+b?eF.$!1F7&)e@]P8F.7$$!1TzQV)Rbw &F.$!1:!)pjIo;9F.7$$!1i$*[NoI'p&F.$!1rE4s]1\"\\\"F.7$$!1KyxTTVe_F.$!1P W1q$>#\\;)[h$zYF.$!1F'zh'H-.>F.7$$!1&GnH9RT2%F.$!1J#=U./e)>F .7$$!18ZpkYw&f$F.$!12r7Pdki?F.7$$!19YPu[94NF.$!1+4,>KD([wxuLF.$!1/.zk3!o$HF.7$$!18'QY@HM_$F.$!1DzqJ]<%[$F.7$$!1+6C$>]ii$F .$!1#)zU?J1KTF.7$$!1Wu)zv)efNF.$!10!3)QJT'o%F.7$$!1P'[$H7U:NF.$!1#*el7 T:%y%F.7$$!1Sk(**3*GgMF.$!1o`(Rc]$p[F.7$$!12EkxIW%R$F.$!1-SIie0T\\F.7$ $!1L+Sv*=$=LF.$!18gt_=`)*\\F.7$$!1kZ)G1CDB$F.$!1wKO^gET]F.7$$!1A(*[m.# y8$F.$!1#H1:y%)*o]F.7$$!139+3^5NIF.$!1Ya\"p%>m\"3&F.7$$!1/q4/))QDHF.$! 1%>64AB&z]F.7$$!1hQo47!\\S#F.$!1yfvQ\\[>\\F.7$$!1)3o\"o)f(o=F.$!1E?S&o @se%F.7$$!15!z!RrVM9F.$!1a+nYIq^UF.7$$!1$G+%G1m!3\"F.$!1Nk\"GyU\"**RF. 7$$!12i'p\\([$)**F1$!1#eI?-ot&RF.7$$!1i:F\"GA`@*F1$!1^G&Gx/,$RF.7$$!1P =t20)f\\)F1$!1]^!p!)p$=RF.7$$!1%\\o*4'>\"=yF1$!1&)o`Un&G#RF.7$$!1'pqwU IM<(F1$!1M(4J(*>R%RF.7$$!1ToCrW!Hb'F1$!1-/'z[l:)RF.7$$!18nE2b6ZfF1$!1G #4\"4)[a.%F.7$$!1IL;g!3lM&F1$!1Cm0;,)[5%F.7$$!1(p6=TMp(HF1$!1R=KjWL%[% F.7$$!1f())QP97_\"Fdhl$!1(z@%y+_v\\F.7$$\"1QXwvg8aMF1$!1[H5)**)HsaF.7$ $\"1*H^k4&fWvF1$!1tt#e$R35eF.7$$\"1)G86Yvtc)F1$!1DMV7yObeF.7$$\"1[!f?t G@e*F1$!1sSM#eSP)eF.7$$\"19MByT*z0\"F.$!17%fa&3d%*eF.7$$\"1G&3W$))>b6F .$!193ktC_()eF.7$$\"1lZVmm(*[7F.$!1*ez_UfD'eF.7$$\"1^&)>^K`Q8F.$!1&*Q] w]%*>eF.7$$\"1E?,)=_JU\"F.$!15)*ybdBgdF.7$$\"1#>MFz:A]\"F.$!13STR)oUo& F.7$$\"1z$Rc\\JUw\"F.$!1@GC5`L=_F.7$$\"1ia(oB@\\\">F.$!1,cYi)o.h%F.7$$ \"1rz&yrb4*>F.$!1nJhS+RISF.7$$\"1/<(4%4fn?F.$!16:p4B-yNF.7$$\"1)y'[W,& o4#F.$!1sNWAL-$\\$F.7$$\"1d-K>OaK@F.$!16!\\R)3*>U$F.7$$\"1aEZ\"exd<#F. $!1\"[DxYh^O$F.7$$\"1**fkBk^FAF.$!1*\\\"pD#oCK$F.7$$\"1I)yn`b&)G#F.$!1 os*GbZNH$F.7$$\"1tR*[S(\\fBF.$!1?Ub*o^xF$F.7$$\"1>><'eJ2W#F.$!1:@.@q;u KF.7$$\"1J!Q?6>C`#F.$!1,.&zhP;G$F.7$$\"1grJ'RES*HF.$!13z#R*3!4R$F.7$$ \"1$e_]1oBd$F.$!1j*)e]y*Qa$F.7$$\"1IYhy#o'GTF.$!1xfj'ywgi$F.7$$\"1)H#p PWO;YF.$!1$eQPr[$F.7$$\"16>L\\YP;\\F.$!1HGT!y8)>MF.7$$\"1o0bVMP%)\\F.$!1OO m5K'*RLF.7$$\"1vDeG#G[.&F.$!1)\\o1O&H[KF.7$$\"1!y.l@3t1&F.$!1%>GDtnd9$ F.7$$\"1\"R6<2P<3&F.$!12.Chv`LIF.7$$\"1)Q$z\\/Sy]F.$!1U)GWuFH\"HF.7$$ \"1%yF.7$$\"1?$eY HQ\\F%F.$!1a^#*Hz!RY\"F.7$$\"1>/ZAA5)*RF.$!1Z^,9C$)y5F.7$$\"186OZD`eRF .$!1Zsfi$**4+\"F.7$$\"1eL\"fS&*=$RF.$!1Sd`:P$4G*F17$$\"1C<0Bd1>RF.$!1= 84TM.'f)F17$$\"1SaCkE0L(>\\zF17$$\"1(R;\"=C.PRF.$!1Z^]6)GNL (F17$$\"1&)z;m$\\#oRF.$!1BI$3gH:u'F17$$\"11S82V59SF.$!1oDnN.JlhF17$$\" 1oW$yI6T2%F.$!1e#eo[2of&F17$$\"1%QV(=EAqWF.$!1@4\"Q?Jh0$F17$$\"15Lc1++ +]F.$\"1`7pv&o?5%!#C-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%%VIEWG6$%(DEFAUL TGFffn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here is the plot of " } {XPPEDIT 18 0 "r = 2*cos(theta);" "6#/%\"rG*&\"\"#\"\"\"-%$cosG6#%&the taGF'" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 11 " from 0 to " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 29 ". Note that in this example " }{TEXT 299 2 "r " }{TEXT -1 17 " is ne gative for " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 9 " betwee n " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 22 ", and these values o f " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 53 " correspond to \+ the bottom half of the circle you see." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "polarplot(2*cos(theta ), theta=0..Pi,scaling = constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"#\"\"!F*7$$\"14hRPij!* >!#:$\"1kwb#=y_O\"!#;7$$\"16B))4-Qn>F.$\"1C\\ff*)GLDF17$$\"1/h)yke[#>F .$\"1Mj&[K5J!QF17$$\"1'oz=42`'=F.$\"1:NXR4U7]F17$$\"10vG._U!z\"F.$\"1a H-qceDhF17$$\"1NR'oTd#3jubk6;F.$\"1:7_\\!>8 \"zF17$$\"1V2jbr2,:F.$\"1$>p(Qh-a')F17$$\"1#3nW'R,#Q\"F.$\"1L+16bcT#*F 17$$\"1Z@iF'QDD\"F.$\"1&Q\"[M\"oen*F17$$\"17:E#o8X8\"F.$\"1*fVPO<\"4** F17$$\"1efAI*GS***F1$\"08;t@)******F.7$$\"1/&e2kwGZn!**F1 7$$\"1XGk\\y\"QN(F1$\"1A>=\\D`V'*F17$$\"1Ac$*o5*oA'F1$\"1!yAe`m3E*F17$ $\"1.vPEZod\\F1$\"1V'yI2&oN')F17$$\"1QRkB0^gRF1$\"1i\")RQ+BqzF17$$\"1c dB`Oo5HF1$\"1fvOg?x_qF17$$\"1=uW%z&z&4#F1$\"14(*=gKpS 8F1$\"1$H$\\4/k,]F17$$\"1vajosj!p(!#<$\"1n'[*>HvXQF17$$\"1F4'f!\\wjLF^ r$\"1%p9m#Q%=d#F17$$\"1$H-$e!)*pH*!#=$\"1V?00]Ug8F17$$\"1UI-u;24[!#@$ \"1DA\\O%485$Fir7$$\"1Q[pF\")>c\"*Fir$!1$4*>c=8]8F17$$\"1\"[kuKX$fKF^r $!1NbFGIGKDF17$$\"1s%o'pyPntF^r$!15&=j`Bsw$F17$$\"1Dvq=WL\"R(F1$!1T>(fp[Pl*F17$$\"1)y eZAL%3()F1$!1$RoI*>C;**F17$$\"1>jqlhUp**F1$!11O#>E`*****F17$$\"1BKODBx G6F.$!1YZ3X=u;**F17$$\"1$RqX8/bD\"F.$!1$[o%H'z!o'*F17$$\"1)yb&Q)zNQ\"F .$!1/2<#>x]B*F17$$\"1h(4w0I.]\"F.$!1Hgb5wMe')F17$$\"1D)[*R]$4h\"F.$!1a *[=H2o\"zF17$$\"1gnkQ+$*4F.$!1'>2ndo*fQF17$$ \"1(GAp))oa'>F.$!1g]nRQ=0EF17$$\"1elf`bp!*>F.$!1up91t'4O\"F17$F($\"1YK hSr8/#)!#D-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG- %%VIEWG6$%(DEFAULTGFg[l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "To se e that " }{XPPEDIT 18 0 "r = 2*cos(theta)" "6#/%\"rG*&\"\"#\"\"\"-%$co sG6#%&thetaGF'" }{TEXT -1 182 " really does give the circle with cen ter at (1,0) and radius 1, as it appears to, we can proceed as follows . Recall that the equation of this circle in rectangular coordinates \+ is " }{XPPEDIT 18 0 "(x-1)^2+y^2-1 = 0;" "6#/,(*$,&%\"xG\"\"\"F(!\"\" \"\"#F(*$%\"yGF*F(F(F)\"\"!" }{TEXT -1 74 ", and that polar and rectan gular coordinates are related by the equations " }{XPPEDIT 18 0 "x = r *cos(theta);" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 7 " , and " }{XPPEDIT 18 0 "y = r*sin(theta);" "6#/%\"yG*&%\"rG\"\"\"-%$s inG6#%&thetaGF'" }{TEXT -1 108 ". Substitute the last two equations i nto the first equation and simplify. Let's have Maple do the algebra. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "subs(x=r*cos(theta),y=r*sin(theta), (x-1)^2+y^2-1=0); expand(%); simplify(%/r); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$), &*&%\"rG\"\"\"-%$cosG6#%&thetaGF*F*!\"\"F*\"\"#F*F**&)F)F0F*)-%$sinGF- F0F*F*F/F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"rG\"\"#\" \"\")-%$cosG6#%&thetaGF(F)F)*&F'F)F+F)!\"#*&F&F))-%$sinGF-F(F)F)\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$cosG6#%&thetaG!\"#%\"rG\"\" \"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "The polar equation is on the last line. What do you think " } {XPPEDIT 18 0 "r = 2*sin(theta);" "6#/%\"rG*&\"\"#\"\"\"-%$sinG6#%&the taGF'" }{TEXT -1 24 " looks like? How about " }{XPPEDIT 18 0 "r = -9* cos(theta);" "6#/%\"rG,$*&\"\"*\"\"\"-%$cosG6#%&thetaGF(!\"\"" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The polar equations lines through the origin have simple equation s, name, " }{XPPEDIT 18 0 "theta = c;" "6#/%&thetaG%\"cG" }{TEXT -1 7 " where " }{TEXT 300 4 " c " }{TEXT -1 156 "is a constant. For ot her lines we can find the polar equation by first writing an equation for the line in rectangular coordinates and then substituting " } {XPPEDIT 18 0 "x = r*cos(theta);" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&the taGF'" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "y = r*sin(theta);" "6#/% \"yG*&%\"rG\"\"\"-%$sinG6#%&thetaGF'" }{TEXT -1 58 ". For example, th e vertical line x=5 has polar equation " }{XPPEDIT 302 0 "r*cos(theta ) = 5;" "6#/*&%\"rG\"\"\"-%$cosG6#%&thetaGF&\"\"&" }{TEXT -1 5 " or \+ " }{XPPEDIT 18 0 "r = 5/cos(theta);" "6#/%\"rG*&\"\"&\"\"\"-%$cosG6#%& thetaG!\"\"" }{TEXT -1 34 ". What do you expect the command " }{TEXT 301 42 "polarplot(3/sin(theta), theta=Pi/4..Pi/2);" }{TEXT -1 12 " to \+ produce?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "END OF HIDDEN SECTION. Scroll up to close. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 330 10 "Example 3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "drdtheta(theta= 0..Pi/2, r=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&-%\"FG6$%\"rG%&theta G\"\"\"F,F./F,;F.\"\"#/F-;\"\"!,$%#PiG#F.F1" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6<-%'CURVESG6$7S7$$\"\"\"\"\"!F*7$$ \"1H!eZ1RT***!#;$\"1R6)y.?KU$!#<7$$\"1')>%3#y]z**F.$\"1f:jY@i)R'F17$$ \"1#)=0ATZ_**F.$\"1g.'oLeyt*F17$$\"1Q.%\\wzR\"**F.$\"1A'pLD=)38F.7$$\" 1$>q:^tW')*F.$\"17`5:5yS;F.7$$\"1[vT(GY'3)*F.$\"1_xPh)4p%>F.7$$\"1l#p[ UK3u*F.$\"1/Q(\\C)*=E#F.7$$\"1.WrxG1g'*F.$\"1hqR=d=&e#F.7$$\"1LZ.4u()o &*F.$\"1g)*eE+e/HF.7$$\"1[H,W$oSY*F.$\"1>43!f5]F.7$$\"18\"p)Q<++&)F.$\"1[2kqS#yE&F. 7$$\"1Tfb,`56$)F.$\"1/)R$=E2hbF.7$$\"1K/mF.$\"1 YH8\")o&)3vF.7$$\"1`(Qk;xjN'F.$\"1_Z\"[&\\()>xF.7$$\"1'Q3.v_J4'F.$\"1 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0 "theta;" " 6#%&thetaG" }{TEXT -1 7 "\" or \"d" }{XPPEDIT 18 0 "theta;" "6#%&theta G" }{TEXT -1 53 "dr\" type, or both if specified, for an function F(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 "). Use the" } {TEXT 257 9 "drdtheta " }{TEXT -1 4 "and " }{TEXT 258 8 "dthetadr" } {TEXT -1 1 " " }{TEXT 259 8 "commands" }{TEXT -1 36 " to experiment an d check your work. 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