Surface Area IntegralsMichael Cook & Robert GarayMath 253 - Honors Spring 2009restart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):with(VecCalc):VCalias:
StartEngine():randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):randm:=rand(1..5):randn:=rand(1..2):randrmax:=rand(1..4):shape[1]:="cone":shape[2]:="paraboloid":initplot:=plot3d([r*cos(theta),r*sin(theta),0],r=0..2,theta=0..2*Pi, grid=[10,49], axes=normal, tickmarks=[0,0,0], orientation=[30,75], color=white):problem:=proc()
global m,n,rmax,r_max,R,initplot,currplot,e_r,e_theta,N,lN,SurfArea, valSurfArea, r, theta:
m:=randm():
n:=randn():
rmax:=randrmax():
r_max:="":
R:=[r*cos(theta),r*sin(theta),m*r^n]:
currplot:=plot3d(R,r=0..rmax,theta=0..2*Pi, grid=[10,49], scaling=constrained, axes=normal, tickmarks=[0,0,0], orientation=[30,75]):
e_r:=diff(R,r):
e_theta:=diff(R,theta):
N:=Cross(e_r,e_theta):
lN:=Length(N) assuming r>0:
SurfArea:=Muint(lN,r=0..rmax,theta=0..2*Pi):
valSurfArea:=simplify(value(SurfArea)):
Set(probstate=sprintf(" Find the surface area of the %s given by %a below %a.\134n The surface can be parametrized as: %a.", shape[n], z=m*(x^2+y^2)^(n/2), z=m*rmax^n, R)):
Set(surfplot=currplot):
Set(TFrmax=""):
Set(TFe_r=""):
Set(TFe_theta=""):
Set(TFN=""):
Set(TFlN=""):
Set(TFSurfArea=""):
end proc:hint1:=proc()
Set(reply=sprintf("%a",'z_max'=m*'r_max'^n)):
#Set(reply=cat("",convert(z_max=m,string),"*r_max^",convert(n,string),".")):
end proc:check1:=proc()
global rmax:
local user_rmax:
if Get(TFrmax)="" then
Set(reply="No answer given."):
return:
end if:
user_rmax:=Get(TFrmax::anything):
if simplify(rmax - user_rmax)=0 then
Set(reply="Radical Dude!"):
else
Set(reply="I thought you were better than that."):
end if:
end proc:show1:=proc()
global rmax:
Set(TFrmax=rmax):
end proc:hint2:=proc()
Set(reply="Find the Partial Derivative with respect to r."):
end proc:check2:=proc()
global e_r:
local user_e_r:
if Get(TFe_r)="" then
Set(reply="No answer given."):
return:
end if:
user_e_r:=Get(TFe_r::list, corrections=true, update=true):
if simplify(e_r - user_e_r)=[0,0,0] then
Set(reply="You're the Bomb!"):
else
Set(reply="Are you sure you're good at math?."):
end if:
end proc:show2:=proc()
global e_r:
Set(TFe_r=e_r):
end proc:hint3:=proc()
Set(reply="Find the Partial Derivative with respect to theta."):
end proc:check3:=proc()
global e_theta:
local user_e_theta:
if Get(TFe_theta)=""then
Set(reply="No answer given."):
return:
end if:
user_e_theta:=Get(TFe_theta::list, corrections=true, update=true):
if simplify(e_theta - user_e_theta)=[0,0,0] then
Set(reply="You Rock!"):
else
Set(reply="Oops...try again."):
end if:
end proc:show3:=proc()
global e_theta:
Set(TFe_theta=e_theta):
end proc:hint4:=proc()
Set(reply="Find the Cross Product of e_r and e_theta."):
end proc:check4:=proc()
global N:
local user_N:
if Get(TFN)=""then
Set(reply="No answer given."):
return:
end if:
user_N:=Get(TFN::list, corrections=true, update=true):
if simplify(N - user_N)=[0,0,0] then
Set(reply="Who's got the skills...You!"):
else
Set(reply="Better luck next time."):
end if:
end proc:show4:=proc()
global N:
Set(TFN=N):
end proc:hint5:=proc()
Set(reply="Take the square root of the sum of the squares of the components of N."):
end proc:check5:=proc()
global lN:
local user_lN:
if Get(TFlN)="" then
Set(reply="No answer given."):
return:
end if:
user_lN:=Get(TFlN::anything):
if simplify(evalf(lN) - evalf(user_lN))=0 then
Set(reply="Way to go!"):
else
Set(reply="Nope, keep trying."):
end if:
end proc:show5:=proc()
global lN:
Set(TFlN=lN):
end proc:hint6:=proc()
Set(reply="Use a double integral with respect to r and theta"):
end proc:check6:=proc()
global valSurfArea:
local user_SurfArea:
if Get(TFSurfArea)="" then
Set(reply="No answer given."):
return:
end if:
user_SurfArea:=Get(TFSurfArea::anything):
if simplify(valSurfArea - value(user_SurfArea))=0 then
Set(reply="Great Job! Now try another problem."):
else
Set(reply="You'll do better next time for sure."):
end if:
end proc:show6:=proc()
global valSurfArea:
Set(TFSurfArea=valSurfArea):
end proc:SurfacArea:=Maplet(onstartup=RunWindow(MAIN),
Font[helv10]("helvetica",10),
Window[MAIN](title="Surface Area of Cones and Parabaloids",
[ halign=none,
[ Button("New Problem", Evaluate(function=problem), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ inset=0,
TextBox[probstate](sprintf(" Find the surface area of the ... given by ... below ... .\134n The surface can be parametrized as: ... ."), width=75, height=2)#,
],
[ [ border=true,
caption="Rotate the plot with the mouse.",
Plotter[surfplot](initplot, width=200,height=300)
],
[ inset=0,
[ border=true,
caption="Find the maximum value of r:",
"r_max =",
TextField[TFrmax](width=5, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint1"), background=lightorange),
Button("Check", Evaluate(function="check1"), background=green),
Button("Show", Evaluate(function="show1"), background=lightorange)
],
BoxColumn( border=true,
caption="Find the partial derivatives of the equation with respect to r and theta:",
[ inset=0,
"e_r = ",
TextField[TFe_r](width=25, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint2"), background=lightorange),
Button("Check", Evaluate(function="check2"), background=green),
Button("Show", Evaluate(function="show2"), background=lightorange)
],
[ inset=0,
"e_theta =",
TextField[TFe_theta](width=25, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint3"), background=lightorange),
Button("Check", Evaluate(function="check3"), background=green),
Button("Show", Evaluate(function="show3"), background=lightorange)
]
),
[ border=true,
caption="Find the Normal Vector",
"N =",
TextField[TFN](width=30, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint4"), background=lightorange),
Button("Check", Evaluate(function="check4"), background=green),
Button("Show", Evaluate(function="show4"), background=lightorange)
],
[ border=true,
caption="Find the Length of the Normal Vector",
"Length of N =",
TextField[TFlN](width=24, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint5"), background=lightorange),
Button("Check", Evaluate(function="check5"), background=green),
Button("Show", Evaluate(function="show5"), background=lightorange)
],
[ border=true,
caption="Find the Surface Area of the Surface",
"Surface Area =",
TextField[TFSurfArea](width=23, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint6"), background=lightorange),
Button("Check", Evaluate(function="check6"), background=green),
Button("Show", Evaluate(function="show6"), background=lightorange)
]
]
],
[ TextField[reply](width=55, editable=false, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Michael Cook & Robert Garay", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2009", font=FontFooter)
]
]
)):Maplets[Display]( SurfacArea ):