aŌ5 ChangeCoords2* Corners Coords* j k CornersTemp*œ–˜D>0 cylindrical<.Ŧ–˜ nops*,* entries*0 true: .*Ƙ,* seq*,* op*Ŧ–˜Ӗ˜*. ž–˜Æ–˜Ŧ–˜ cos*Ӗ˜*ؖ˜Ŧ–˜Ō–˜Ŧ–˜ sin喘Ŧ–˜Ӗ˜*ؖ˜ۖ˜4Ŧ–˜Ą–˜ spherical<ƘŦ–˜Ŧ–˜Ū–˜ū–˜:֘,*˖˜*,*Ō–˜Ŧ–˜ð–˜Ŧ–˜▘*ö–˜Ŧ–˜Ō–˜Ŧ–˜ð–˜Ŧ–˜ņ–˜—˜Ŧ–˜Ņ–˜ü–˜Ą–˜ Cartesian:˘ evalŧ–˜7—˜*˘œ–˜œ–˜ help/text/intdraw TEXT7* FUNCTION: asu[intdraw] -creates a picture of the region of a two-dimentional integral  CALLING SEQUENCE: intdraw(boundaries); intdraw(boundaries, intdraw options, plot options);o—˜ PARAMETERS: boundaries -two 2D integral boundaries written as  var = Lower Limit..Upper Limit. Outer integral must be the first  argument, then inner integral.  INTDRAW OPTIONS: grid -used to vary number of columns; for polar  coordinates, grid's second argument is used  to vary curvature of columns as well. Numbers  entered into grid may not always correlated to number  of columns shown on graph. Default is grid=[15,15].o—˜ spaces -used to vary spacing between columns or boxes.  Spaces arguments must range from 0 to 1. Default is  spaces=[.3,0].  SYNOPSIS: -for Cartesian coordinates, variables used must be x, and y. -for polar coordinates, variables used must be r, and theta. -scaling (plot option) default in intdraw is CONSTRAINED -labels (plot option) default in intdraw is [x,y] -these defaults were choosen because they produced the nicest and  most informative pictures, they may be changed easily in the plot  window or in the intdraw calling statement (see examples bellow)o—˜ #examples intdraw(x=-1..1, y=-sqrt(1-x^2)..sqrt(1-x^2),spaces =[.5], color = blue,  axes = BOX); intdraw(x=-1..1, y=-sqrt(1-x^2)..sqrt(1-x^2),spaces =[.3,.3],color = yellow); intdraw(r= 0..1, theta=0..2*Pi,grid = [3,25]); o—˜ region:=plot([3,t,t=0..3]),plot([t,3,t=-3..3]),plot([t,-t,t=-3..-sqrt(2)]), plot([2*cos(t),2*sin(t),t=0..3*Pi/4]):o—˜#™˜8™˜o—˜ int1:=intdraw(theta= 0..Pi/4,r=2..3/cos(theta), color = blue): int2:=intdraw(theta= Pi/4..3*Pi/4,r=2..3/sin(theta), color = green): display([int1,int2,region]);o—˜ inta:=intdraw(r=2..3, theta = 0..3/4*Pi, grid = 4,color = green): intb:=intdraw(r=3..3*sqrt(2), theta = arccos(3/r)..arcsin(3/r), grid = [10,25], color = blue): intc:=intdraw(r=3..3*sqrt(2), theta = Pi-arcsin(3/r)..3/4*Pi, grid = 6, color = red): display([inta,intb,intc,region]); MakeCorners2* CentersTable a b c c0 c1 SpaceA SpaceB SpaceC DeltaA DeltaB DeltaC xPlace yPlace zPlace g CSArg * f i ctr u0 u1 v0 v1 w0 w1—–˜œ–˜œ–˜D<ƘŦ–˜Ŧ–˜Ū–˜ū–˜ D:˘Ӗ˜*Ж˜ī–˜:. evalf*Ӗ˜*Ŧ–˜Ä–˜Ŧ–˜Ŧ–˜Ŧ–˜0Ŧ–˜0 Ŧ–˜Wš˜é–˜:.Eš˜*Kš˜Ŧ–˜Qš˜Ŧ–˜é–˜:.Eš˜*Ӗ˜*閘˘Ŧ–˜Ŧ–˜Ŧ–˜0Wš˜Ŧ–˜0 Ŧ–˜]š˜:.Eš˜*tš˜Ŧ–˜zš˜jš˜>$$Ӗ˜*ų–˜0 theta•š˜ phi0 falseD:. subs*Ą–˜aš˜jš˜Bš˜jš˜0‡š˜jš˜nš˜jš˜0:. °š˜*ĩš˜žš˜0>( type*­š˜ constantū–˜Ԛ˜*Įš˜Úš˜ū–˜ ERROR* Second/third boundry errorD:­š˜7—˜*Ӗ˜*ų–˜Ä–˜Ŧ–˜Ŧ–˜Ŧ–˜0 Wš˜Ŧ–˜0 Ŧ–˜]š˜:Įš˜7—˜*ýš˜Ŧ–˜›˜jš˜:ۖ˜,*,*Bš˜aš˜,*nš˜‡š˜,*­š˜Įš˜: . Ɩ˜, *,*Ӗ˜*Ŧ–˜Ӗ˜*0 ۖ˜Ӗ˜*Ŧ–˜Ӗ˜*0ۖ˜Ӗ˜*Ŧ–˜Ӗ˜*0ۖ˜,*2›˜<›˜Ӗ˜*閘J›˜,*2›˜Ӗ˜*閘@›˜F›˜,*2›˜\›˜T›˜,*Ӗ˜*閘6›˜<›˜F›˜,*i›˜<›˜T›˜,*i›˜\›˜F›˜,*i›˜\›˜T›˜7—˜*+›˜œ–˜œ–˜ asu8œ–˜§š˜JH intdraw2œ–˜)*Ų™˜ a0 a1ܙ˜ b0 b1 CountA CountBč™˜ė™˜ TempArgsә˜ CornersTableŒ–˜ TempTable PolygonsTable PlotTitleš˜‘–˜ ggô™˜ø™˜š˜š˜ Axes Scaling Style xTick yTick CountArgs SpaceArgs SpaceBArg CountBArg TitleArgs ScalingArgs kk Options ggg NASpacing NBSpacingœ–˜œ–˜#D:.#§š˜:.§š˜:.§š˜:."§š˜:. §š˜.!:.%,œ–˜:.'§š˜:.(§š˜>"0閘蚘* Boundaries must be entered as first 2 arguments>((((Ԛ˜* 0ĸĸĸĸ*Ŧ–˜ =§š˜Ԛ˜* ۜ˜*é–˜ēœ˜§š˜Ԛ˜*Ӗ˜*閘­œ˜ range§š˜Ԛ˜*Ӗ˜*閘šœ˜Éœ˜§š˜Îœ˜蚘* Boundaries must be written as var=lower limit..upper limit:ۖ˜Ӗ˜*Ŧ–˜­œ˜:ƘEš˜*Ӗ˜*Ŧ–˜Äœ˜:˘Eš˜*Ӗ˜*閘Ĝ˜:Bš˜Ӗ˜*Ŧ–˜šœ˜:aš˜Eš˜*Ӗ˜*Ŧ–˜Ōœ˜:nš˜Eš˜*Ӗ˜*é–˜Ōœ˜>"閘œ˜<.$ų–˜Ŧ–˜œ˜ū–˜ >Ӗ˜*Ŧ–˜ ۜ˜*)˜ gridD:pœ˜ū–˜:‡š˜ round*Eš˜*Ӗ˜*Ŧ–˜Ӗ˜*閘5˜>"Ŧ–˜Ŋ–˜*P˜D:œ˜ū–˜:­š˜D˜*Eš˜*Ӗ˜*閘P˜:­š˜1˜ spacesD:tœ˜ū–˜:Įš˜I˜>V˜D:|œ˜ū–˜:+›˜e˜:+›˜ų–˜Wš˜1˜ titleD:xœ˜ū–˜:.P˜1˜ scalingD:lœ˜ū–˜:.P˜:‚œ˜,*Ӗ˜*‚œ˜5˜>xœ˜§š˜:—˜ cat*ۖ˜o—˜Bš˜>lœ˜§š˜:Ĩ˜ CONSTRAINED>pœ˜§š˜D:‡š˜o˜m˜>tœ˜§š˜D:Įš˜‰˜:+›˜>((("Įš˜Ý˜"Ŧ–˜Įš˜"+›˜Ý˜"Ŧ–˜+›˜蚘* spaces arguments must be between or equal to 0 and 1:.&,*ۖ˜Bš˜ß™˜>$ member* xž˜ū–˜ž˜* yž˜ū–˜D:./—˜>(ž˜ž˜ž˜ž˜蚘* x, and y are key variables and must be cleared before running program, (ex x:='x')ž˜*ž˜ž˜.ž˜*ž˜ž˜.$ž˜* rž˜ū–˜ž˜*›š˜ž˜ū–˜D:#ž˜Ģ–˜ž˜*až˜ž˜Qž˜ž˜*›š˜ž˜Yž˜>(až˜až˜›š˜›š˜蚘* r, and theta are key variables and must be cleared before running program, (ex r:='r')>$Zž˜Ŧ–˜Eš˜*˘Ŧ–˜Đ–˜Wš˜Eš˜* Pi閘:‡œ˜ū–˜>$Zž˜é–˜Eš˜*nš˜Ŧ–˜aš˜Wš˜ēž˜:‹œ˜ū–˜蚘* Cartesian coords accepts only x, and y variables polar coords accepts only r, and theta variables:. MakeGrid2D*ۖ˜Đ–˜Ä–˜Bš˜aš˜nš˜‡š˜­š˜Įš˜+›˜#ž˜ž˜œ˜|œ˜:.  ņž˜°œ˜:. ņž˜žœ˜:. ņž˜*ų–˜:.  MakeCorners2D* Ÿ˜Û–˜Bš˜aš˜nš˜Įš˜+›˜Ÿ˜Ÿ˜Rž˜Zž˜ž˜œ˜|œ˜:Ÿ˜ ChangeCoords2D*Ÿ˜#ž˜:. MakePoly2D*Ÿ˜ Ÿ˜ with* plots polygonplot*6*˖˜* AŸ˜*.eŸ˜4Ŧ–˜Ŋ–˜*,*·–˜* Ÿ˜Ž˜—˜œ˜Ĩ˜Ž˜œ–˜œ–˜H intdraw3d2œ–˜8*Ų™˜å›˜č›˜Ü™˜ė›˜ï›˜ß™˜â™˜å™˜ō›˜ö›˜ CountCč™˜ė™˜ð™˜ü›˜Ó™˜œ˜Œ–˜ œ˜œ˜œ˜š˜‘–˜œ˜ô™˜ø™˜ü™˜š˜š˜š˜"œ˜&œ˜*œ˜.œ˜2œ˜ zTick6œ˜;œ˜Jœ˜Oœ˜Tœ˜Wœ˜[œ˜^œ˜cœ˜ NCSpacingš˜ Labels LabelsArgs"œ˜ AxesArgs aTemp bTemp cTempœ–˜œ–˜,D:.)§š˜:ž˜§š˜†œ˜:.4§š˜Šœ˜:.+„œ˜:.-§š˜:..§š˜:./§š˜:.0§š˜:.2§š˜>"œ˜ų–˜蚘* Boundaries must be entered as first 3 arguments>((((Ϝ˜Ԛ˜* ۜ˜Ÿ˜ēœ˜§š˜Āœ˜Îœ˜Îœ˜蚘* Boundaries must be written as var=rangeõœ˜ĸœ˜ ˜˜˜:‡š˜Ӗ˜*Ŧ–˜  ˜:­š˜Eš˜*Ӗ˜*Ŧ–˜Ӗ˜*閘  ˜:Įš˜Eš˜*Ӗ˜*閘͠˜>"ų–˜œ˜<.*Ŧ–˜œ˜ū–˜>Ӗ˜*Ŧ–˜ ۜ˜*᠘9˜D:ž˜ū–˜:+›˜Eš˜*Ӗ˜*Ŧ–˜Ӗ˜*閘>"Ŧ–˜Ŋ–˜*Ą˜:. Eš˜*Ӗ˜*閘Ą˜: Ą˜>"閘Ą˜D: Ÿ˜Eš˜*Ӗ˜*ų–˜Ą˜:z ˜ū–˜: Ÿ˜Ą˜ꠘs˜Dšž˜>ü ˜ slicesD:Ÿ˜—˜Wš˜:#ž˜Ŧ–˜Wš˜:ņž˜Ý˜D:Ÿ˜ų ˜>Ą˜:#ž˜Ą˜:#ž˜‰˜>Ą˜D:ņž˜"Ą˜*Ą˜GĄ˜ꠘŽ˜DĖž˜:Ÿ˜Ą˜ꠘœ˜D:] ˜ū–˜:œ˜Ą˜ꠘ axesD:e ˜ū–˜:|œ˜Ą˜ꠘ labelsD:~ ˜ū–˜:.1Ą˜:j ˜,*Ӗ˜*j ˜î ˜>~ ˜§š˜:ˆĄ˜,*ž˜ž˜ z>e ˜§š˜:|œ˜ FRAME>‹œ˜§š˜:Ÿ˜ļ˜*ۖ˜o—˜Bš˜o—˜‡š˜>] ˜§š˜:œ˜Å˜>ž˜§š˜D:+›˜Ą˜Ą˜-Ą˜>‡œ˜§š˜D:Ÿ˜‰˜SĄ˜GĄ˜>((((("Ÿ˜Ý˜"Ŧ–˜Ÿ˜"#ž˜Ý˜"Ŧ–˜#ž˜"ņž˜Ý˜"Ŧ–˜ņž˜蚘* spaces arguments must range from 0 and 1>(( +›˜Ý˜ Ą˜Ý˜ Ÿ˜Ý˜蚘* grid arguments must be integers that are greater than 0:.,,*ۖ˜Bš˜‡š˜>$$ž˜*ž˜ǘū–˜ž˜*ž˜ǘū–˜ž˜*žĄ˜ǘū–˜D:./—˜>('ž˜žĄ˜žĄ˜蚘* x, y, and z are key variables and must be?ž˜ž˜*ž˜ǘ.ž˜*ž˜ǘpœ˜ž˜*žĄ˜ǘtœ˜$$ž˜*až˜ǘū–˜ž˜*›š˜ǘū–˜4ǘ D:=Ē˜Ģ–˜>((ƒž˜Cǘž˜蚘* z, theta, and r are key variables and must be—ž˜ž˜*až˜ǘ]ǘž˜*›š˜ǘeǘgǘ>$pœ˜Ŧ–˜Đž˜:n ˜ū–˜>$pœ˜é–˜Âž˜:r ˜ū–˜>$pœ˜ų–˜Eš˜*Įš˜Ŧ–˜­š˜Wš˜ēž˜:v ˜ū–˜$$ž˜* rhoǘū–˜ž˜*Ąš˜ǘū–˜wǘ D:=ǘ—˜ž˜*Ōǘǘ]ǘž˜*Ąš˜ǘeǘž˜*›š˜ǘlǘ>((ƒž˜ŌǘŌǘĄš˜Ąš˜蚘* theta, rho, and phi are key variables and must be cleared before running program, (ex rho:='rho')>$(ŠĒ˜tœ˜Ŧ–˜Đž˜Ūǘ>$(ģǘtœ˜é–˜Âž˜·Ē˜>$(žĒ˜tœ˜ų–˜ŋǘÉǘ蚘* Cartesian coords accepts only x, y, and z variables cylindrical coords accepts only r, theta, and z variables spherical coords accepts only rho, theta, and phi variables:. MakeGrid*ۖ˜Đ–˜Ä–˜Bš˜aš˜nš˜‡š˜­š˜Įš˜+›˜ Ą˜ Ÿ˜Ÿ˜#ž˜ņž˜=ǘǘn ˜r ˜v ˜z ˜:—˜ q̘°œ˜:Ĩ˜ q̘žœ˜:. q̘Ÿ˜:. q̘*㠘:eŸ˜͙˜*—˜Û–˜Bš˜‡š˜­š˜Įš˜Ÿ˜#ž˜ņž˜Ĩ˜šĢ˜ Ģ˜^ǘpœ˜tœ˜ǘz ˜:eŸ˜–˜*eŸ˜=ǘ:Ÿ˜ MakePoly*eŸ˜—˜LŸ˜ polygonplot3d*6*˖˜*˖˜* Ÿ˜*Rž˜Zž˜Zž˜4Ŧ–˜Rž˜4Ŧ–˜Ŋ–˜*,*·–˜*—˜Ž˜Ÿ˜œ˜œ˜rĄ˜|œ˜Ą˜ˆĄ˜Ą˜œ–˜œ–˜ ChangeCoords2D2‡–˜–˜œ–˜œ–˜D> –˜<ƘŦ–˜Ŧ–˜Ū–˜ū–˜:֘,*˖˜*,*Ņ–˜í–˜ۖ˜4Ŧ–˜ã ˜-—˜4—˜<—˜œ–˜œ–˜ MakeGrid2D2*Ų™˜å›˜č›˜Ü™˜ė›˜ï›˜ō›˜ö›˜č™˜ė™˜Œ–˜ š˜Eœ˜@œ˜ NASpaceing NBSpaceing *š˜”–˜‘–˜ aTable bTable b0Table b1Table bMin bMax CenterGridô™˜ø™˜œ–˜œ–˜D>"Eš˜*―š˜Eš˜*Ą–˜蚘* lower limit exceeds upper limit>ŊΘŦΘ蚘* lower limit equal to upper limit>(^œ˜ū–˜Uš˜Ŧ–˜: Ą˜Eš˜*―š˜Ŧ–˜Ą–˜Wš˜Ŧ–˜Uš˜Wš˜: Ą˜Eš˜*āΘŦ–˜Uš˜Ŧ–˜›˜Wš˜Wš˜<ƘŦ–˜Ŧ–˜Uš˜ū–˜: Bš˜Æ–˜Eš˜*Ą–˜Ŧ–˜ƘŦ–˜›˜]š˜Ŧ–˜ Ą˜Ŧ–˜Ŧ–˜ Ą˜]š˜>(Ԛ˜*Eš˜*°š˜*ž–˜ Bš˜°œ˜Äš˜Úš˜ū–˜Ԛ˜*Eš˜*°š˜*Ĩ˜Îš˜Úš˜ū–˜蚘* invalid boundaries<ƘŦ–˜Ŧ–˜Uš˜ū–˜D: nš˜Æ–˜Eš˜*°š˜*ž–˜ýΘ̘: ‡š˜Æ–˜Eš˜*°š˜*OĨ˜Îš˜:Įš˜ max*˖˜*Ӗ˜*Ӗ˜*Ƙ,*·–˜*‡š˜Ƙ4Ŧ–˜Ŋ–˜*pĨ˜Eš˜*°š˜*ž–˜Ą–˜Îš˜Eš˜*°š˜*ž–˜―š˜Îš˜:­š˜ min*˖˜*Ӗ˜*Ӗ˜*Ƙ,*·–˜*nš˜Ƙ4Ŧ–˜Ŋ–˜*ĢĨ˜Eš˜*°š˜*„Ĩ˜Äš˜Eš˜*°š˜*ŽĨ˜Äš˜>"Įš˜­š˜ģΘ>(cœ˜ū–˜~š˜Ŧ–˜: Ÿ˜Eš˜*ÃǘŦ–˜~š˜Wš˜: Ÿ˜Eš˜*ÃǘŦ–˜~š˜Ŧ–˜Zš˜Wš˜Wš˜<˘Ŧ–˜Ŧ–˜~š˜ū–˜: aš˜>—˜Eš˜*­š˜Ŧ–˜˘Ŧ–˜Zš˜]š˜Ŧ–˜ Ÿ˜Ŧ–˜Ŧ–˜ Ÿ˜]š˜>((Ӗ˜*閘 ›˜›š˜C›˜ū–˜9›˜ū–˜<ƘŦ–˜Ŧ–˜Uš˜ū–˜<˘Ŧ–˜Ŧ–˜~š˜ū–˜>$"FĨ˜íĨ˜"íĨ˜TĨ˜: +›˜*Ж˜Ä–˜*ýΘíĨ˜<ƘŦ–˜Ŧ–˜Uš˜ū–˜:*›˜ýΘ>›˜+›˜4Ĩ˜*›˜ Ą˜ Ÿ˜œ–˜œ–˜ help/text/intdraw3dI—˜W* FUNCTION: asu[intdraw3d] -creates a picture of a parameterized  three-dimensional region that is sliced according to the selected parameterizationo—˜r—˜ intdraw3d(boundaries); intdraw3d(boundaries, intdraw3d options, plot3d options);o—˜”—˜ boundaries - three ranges in the form u=u0..u1,v=v0..v1,w=w0..w1 u0 and u1 must be constants, v0 and v1 must be expressions  depending on the variable u only, w0 and w1 are expressions that may depend on the variables u and v.  The variables u,v,w may represent either rectangular, cylindrical or spherical coordinates, see below. ԗ˜ INTDRAW3D OPTIONS: grid - determines the numbers of boxes to be drawn. The call grid=[m,n] is interpreted as grid=[m,n,1], i.e. drawing columns, unless the  variable w is either theta or phi, in which case it is interpreted as  grid=[m,n,5]. The default is grid=[5,5,5]. o—˜ spaces - determines the spacing between the boxes (columns).  Its argument is a list of three constants between 0 and 1.  Default is spaces=[.3,.3,0]. The option spaces = slices is  equivalent to spaces= [.8,.1,0]. o—˜[˜˜Ī˜˜ most informative pictures, they may be changed easily in the plot3d  window or in the intdraw3d calling statement (see examples bellow) -for Cartesian coordinates, variables used must be x, y, and z. -for cylindrical coordinates, variables used must be r, theta and z. -for spherical coordinates, variables used must be rho, phi, and theta. -scaling (plot3d option) default in intdraw3d is CONSTRAINED -labels (plot3d option) default in intdraw3d is [x,y,z] -axes (plot3d option) default in intdraw3d is FRAMEo—˜ #if you run all examples at once your computer will be working for a while #Cartesian coordinates examples (wedges) intdraw3d(x= -1..0 , y= -sqrt(1-x^2) ..sqrt(1-x^2) , z= -x..0 ); intdraw3d(z= -1..0 , y= -sqrt(1-z^2) ..sqrt(1-z^2) , x= -z..0 , spaces=[.5,0], axes=BOX); intdraw3d(z= -1..1,x= -sqrt(1-z^2)..sqrt(1-z^2),y= -sqrt(1-x^2-z^2).. sqrt(1-z^2-x^2),spaces=[0,0], style = PATCHNOGRID);o—˜ #cylindrical coordinates examples intdraw3d(theta = 0..2*Pi, r=0..1/sqrt(2), z=r..sqrt(1-r^2), spaces=[0,0]); intdraw3d(z= 0..Pi, r=0..Pi, theta=0..Pi, grid=[3,2], orientation = [-70,60],  spaces=[.4,.4]); intdraw3d(r=3..5, theta = 0..2*Pi, z=-sqrt(1-(4-r)^2).. sqrt(1-(4-r)^2), grid = [5,10 ], axes = BOX );o—˜ I1:= intdraw3d(r=0..3, theta = 0..3*Pi/4, z=-1..1, spaces=[.4,.4],grid = 4, color = green, style = patchnogrid, light = [0,50 , .9,.9,.9],  light = [50,0,.6,.6,.6], ambientlight= [.4,.4,.4]): I2:= intdraw3d(r=3..3*sqrt(2), z=-1..1, theta = arccos(3/r)..arcsin(3/r), grid=[5,3,8],spaces=[.4,.4],color = blue, style = patchnogrid): I3:= intdraw3d(r=3..3*sqrt(2),z=-1..1, theta=Pi-arcsin(3/r)..3/4*Pi, spaces=[.4,.4],grid = [5,3,6], color = red, style = patchnogrid): display3d({I1,I2,I3}, orientation = [-70,50]);o—˜ #spherical coordinates examples intdraw3d(rho=0..1, phi = 0..Pi, theta=0..2*Pi,grid=[1],spaces=[.5,.5]); intdraw3d(rho=0..1,theta= -Pi/2..Pi/2,phi=arctan(1/cos(theta))..Pi/2, grid=[4,6,12]); # changing the spacingÛИ grid=[4,6,12],spaces=[.9,0]); intdraw3d(rho=0..1, theta = 0..2*Pi, phi=0..Pi,grid=[1,8,8], spaces=[0,0]);o—˜ #movie examples (may take a while) with(plots):display([seq(intdraw3d(x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),  z=-sqrt(1-x^2-y^2)..sqrt(1-x^2-y^2),grid=[k,k],spaces =[0,0] ),k=1..10)], insequence = true);o—˜ #this one takes 45 seconds on a Pentium60 int1:=display3d([seq(intdraw3d(x=0..1 , y=-sqrt(1-x^2) ..sqrt(1-x^2) ,  z=0..x , grid = [5,5], spaces=[k/10,0]), k=0..9)], insequence = true): int2:=display3d([seq(intdraw3d(x=0..1 , y=-sqrt(1-x^2) ..sqrt(1-x^2) , z=0..x , grid = [5,5], spaces=[1,k/10]), k=0..9)], insequence = true): int3:=display3d([seq(intdraw3d(x=0..1 , y=-sqrt(1-x^2) ..sqrt(1-x^2) , z=0..x , grid = [5,5], spaces=[1-k/10,1]), k=0..9)], insequence = true): int4:=display3d([seq(intdraw3d(x=0..1 , y=-sqrt(1-x^2) ..sqrt(1-x^2) ,  z=0..x , grid = [5,5], spaces=[0,1-k/10]), k=0.10)], insequence = true): display3d([int1, int2, int3, int4],insequence = true, orientation = [-134,62]); MakePoly2D2*ˆ–˜* TempPolysš˜œ–˜œ–˜D<ƘŦ–˜Ŧ–˜Ū–˜ū–˜: ۖ˜Æ–˜,*Ӗ˜*Ŧ–˜Ý–˜Ӗ˜*閘ݖ˜Ӗ˜*㠘ݖ˜Ӗ˜*ų–˜Ý–˜7—˜*ۖ˜œ–˜œ–˜ MakeGrid2*Ų™˜å›˜č›˜Ü™˜ė›˜ï›˜ß™˜â™˜å™˜ō›˜ö›˜ ˜č™˜ė™˜ð™˜Œ–˜ š˜^œ˜cœ˜8 ˜š˜*š˜”–˜‘–˜‡Ī˜‹Ī˜Ī˜“Ī˜ cTable c0Table c1Table—Ī˜›Ī˜ cMax cMinŸĪ˜ô™˜ø™˜ü™˜œ–˜œ–˜D>ŠĪ˜蚘* lower limit exceeds upper>ÁΘ蚘* lower equal to upper>0ū–˜:AŸ˜Eš˜*āΘŦ–˜Zš˜Wš˜:AŸ˜Eš˜*āΘŦ–˜Zš˜Ŧ–˜9›˜Wš˜Wš˜<ƘŦ–˜Ŧ–˜Zš˜ū–˜:ýΘEš˜*Ą–˜Ŧ–˜ƘŦ–˜9›˜]š˜Ŧ–˜AŸ˜Ŧ–˜Ŧ–˜AŸ˜]š˜>Ĩ˜蚘* invalid first or second boundary<ƘŦ–˜Ŧ–˜Zš˜ū–˜DĨ˜: Ÿ˜aĨ˜: Ą˜”Ĩ˜>" Ÿ˜ Ą˜šŦ˜>0ū–˜:—˜Eš˜* Ÿ˜Ŧ–˜ Ą˜Wš˜Ŧ–˜‚š˜Wš˜:—˜Eš˜*ޘŦ–˜‚š˜Ŧ–˜C›˜Wš˜Wš˜<˘Ŧ–˜Ŧ–˜‚š˜ū–˜:íĨ˜Eš˜* Ą˜Ŧ–˜˘Ŧ–˜C›˜]š˜Ŧ–˜—˜Ŧ–˜Ŧ–˜—˜]š˜>$$Ӗ˜*ų–˜Ĩš˜›š˜KޘĄš˜0§š˜D:eŸ˜Ý˜<ƘŦ–˜Ŧ–˜Zš˜ū–˜<˘Ŧ–˜Ŧ–˜‚š˜ū–˜>͘: ņž˜)͘,͘ D>(Ԛ˜*Eš˜*°š˜*Ĩ˜0 aš˜°œ˜~š˜Úš˜ū–˜Ԛ˜*Eš˜*°š˜*Ĩ˜}ޘ›˜Úš˜ū–˜4Ĩ˜<ƘŦ–˜Ŧ–˜Zš˜ū–˜<˘Ŧ–˜Ŧ–˜‚š˜ū–˜>͘D: Įš˜*Ė˜Đ–˜Eš˜*°š˜*OĨ˜~ޘíĨ˜~š˜: +›˜ĻޘEš˜*°š˜*OĨ˜ēޘ›˜:Ÿ˜bĨ˜*˖˜*Ӗ˜*Ӗ˜*Ƙ,*·–˜›˜Ƙ4Ŧ–˜Ŋ–˜*ŌŽ˜Eš˜*°š˜*„Ĩ˜°š˜*~ޘ̘›˜Eš˜*°š˜*„Ĩ˜°š˜*~Ž˜Îš˜›˜Eš˜*°š˜*ŽĨ˜æŽ˜Eš˜*°š˜*ŽĨ˜ôޘ:#ž˜•Ĩ˜*˖˜*Ӗ˜*Ӗ˜*Ƙ,*·–˜*Įš˜Ƙ4Ŧ–˜Ŋ–˜*­˜Eš˜*°š˜*„Ĩ˜°š˜*鎘~š˜Eš˜*°š˜*ŽĨ˜°š˜*ũޘ~š˜Eš˜*°š˜*ŽĨ˜/­˜4­˜>"Ÿ˜#ž˜šŦ˜>0ū–˜:eŸ˜Ÿ˜Ŧ–˜#ž˜Wš˜Ŧ–˜ ›˜Wš˜:eŸ˜Eš˜*V­˜Ŧ–˜ ›˜Ŧ–˜M›˜Wš˜Wš˜<ƘŦ–˜Ŧ–˜ ›˜ū–˜: ­š˜Æ–˜Eš˜*#ž˜Ŧ–˜ƘŦ–˜M›˜]š˜Ŧ–˜eŸ˜Ŧ–˜Ŧ–˜eŸ˜]š˜<ۖ˜Ŧ–˜Ŧ–˜Zš˜ū–˜<˘Ŧ–˜Ŧ–˜‚š˜ū–˜<ƘŦ–˜Ŧ–˜ ›˜ū–˜>$$$" nš˜TŦ˜íĨ˜"íĨ˜ ‡š˜TŦ˜"Eš˜*°š˜*ž–˜ Bš˜TŦ˜ēޘ~š˜s­˜"s­˜Eš˜*°š˜*Ŋ­˜ēޘ›˜: ņž˜*ۖ˜Ä–˜Đ–˜*୘íĨ˜s­˜>7—˜*ņž˜ņž˜4Ĩ˜*Ï­˜AŸ˜—˜eŸ˜œ–˜œ–˜ MakeCorners2D2*ә˜Ų™˜Ü™˜ė›˜ï›˜č™˜ė™˜ô™˜ø™˜š˜š˜ š˜Eœ˜@œ˜ *š˜š˜š˜š˜ š˜)š˜,š˜—–˜œ–˜œ–˜D<ƘŦ–˜Ŧ–˜Ū–˜ū–˜D:š˜:Bš˜Eš˜*Kš˜Ŧ–˜Ŧ–˜Ŧ–˜Îš˜Wš˜Ŧ–˜~š˜Ŧ–˜]š˜:aš˜Eš˜*Kš˜Ŧ–˜ ۘjš˜>$$͘›š˜9›˜ū–˜C›˜ū–˜D:nš˜°š˜*ĩš˜~ޘ:‡š˜°š˜*ĩš˜Äš˜>(Ԛ˜*nš˜Úš˜ū–˜Ԛ˜*‡š˜Úš˜ū–˜蚘* Second/third boundary errorD:nš˜7—˜*tš˜Ŧ–˜Rš˜Ŧ–˜›˜Ŧ–˜]š˜:‡š˜7—˜*tš˜Ŧ–˜aۘjš˜:ۖ˜,*›˜!›˜:s­˜,*,*Ӗ˜*Ŧ–˜Ӗ˜*Zš˜Û–˜Ӗ˜*閘Ӗ˜*‚š˜Û–˜,*Ӗ˜*閘Ū˜†Ū˜,*}ۘӖ˜*Ŧ–˜ŠŪ˜,*‘Ū˜šŪ˜7—˜*­š˜œ–˜œ–˜ MakePoly2&Ŧ˜(Ŧ˜œ–˜œ–˜D<ƘŦ–˜Ŧ–˜Ū–˜ū–˜D: ۖ˜*ƘŦ–˜<Ŧ˜: ۖ˜*Ж˜é–˜,*Ӗ˜*Ą˜Ý–˜Ӗ˜*čĢ˜Ý–˜Ӗ˜*—˜Ý–˜Ӗ˜*ݖ˜: ۖ˜*Ƙų–˜,*CŦ˜HŦ˜ÓۘÎۘ: ۖ˜*Ж˜ã ˜,*HŦ˜MŦ˜ØŪ˜Óۘ: ۖ˜*ƘĄ˜,*MŦ˜>Ŧ˜ÉۘØŪ˜: ۖ˜*Ƙč̘,*>Ŧ˜CŦ˜ÎۘÉۘRŦ˜œ–˜œ–˜