restart;with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(RandomTools): with(plottools):StartEngine();randomize():randx1:=rand(0..4):
randx2:=rand(1..5):
randtype:=rand(0..1):ask:=proc()
global f, g, a, b, M, Mval, My, Mx, Mxval, Myval, rho, X, Y, intType, u, v, rho1;
local p1, fmin, gmax, rhomin, dv, pad, f1, g1, f2, g2, p2;
rho:=Generate(polynom(integer(range=-3..3), {u,v}, degree=1));
intType:=randtype();
f:=Generate(polynom(integer(range=-3..3), u, degree=2));
g:=Generate(polynom(integer(range=-3..3), u, degree=2));
a:=randx1();
b:=a+randx2();
fmin:=minimize(f, u=a..b);
gmax:=maximize(g, u=a..b);
dv:=fmin-gmax;
if dv<0 then
f:=f-dv;
end if;
rhomin:=minimize(rho, u=a..b, v=minimize(g,u=a..b)..maximize(f,u=a..b));
if rhomin<0 then
rho:=rho-rhomin;
end if;
M:=Int(Int(rho,v=g..f),u=a..b);
if intType=0 then
My:=Int(Int(u*rho,v=g..f),u=a..b);
Mx:=Int(Int(v*rho,v=g..f),u=a..b);else
My:=Int(Int(v*rho,v=g..f),u=a..b);
Mx:=Int(Int(u*rho,v=g..f),u=a..b);
end if;
pad:=(b-a)/5;
if intType=0 then
f1:=subs(u=x,f); g1:=subs(u=x,g);
rho1:=subs({u=x,v=y},rho);
Set(question=cat("Find the center of mass of the region between y = ",
convert(f1,string)," and y = ",convert(g1,string)," for the interval ",a," <= x <= ",
b," with density ",convert(rho1,string),"."));
p1:=densityplot(-rho1,x=a..b,y=g1..f1,color=green);
Set(graph=display(p1, view=[a-pad..b+pad, default]));
else
f1:=subs(u=y,f); g1:=subs(u=y,g);
f2:=subs(u=x,f); g2:=subs(u=x,g);
rho1:=subs({u=y,v=x},rho);
Set(question=cat("Find the center of mass of the region between x = ",
convert(f1,string)," and x = ",convert(g1,string)," for the interval ",a," <= y <= ",
b," with density ",convert(rho1,string),"."));
p2:=densityplot(-rho1, x=a..b, y=g2..f2, color=green);
p1:=reflect(p2, [[0,0], [1,1]]);
p1:=subs(PLOT3D = PLOT, p1);
Set(graph=display(p1, view=[default, a-pad..b+pad]));
end if;
Mval:=value(M);
Myval:=value(My);
Mxval:=value(Mx);
X:=value(My/M);
Y:=value(Mx/M);
Set(MInt="");
Set(MVal="");
Set(MxInt="");
Set(MxVal="");
Set(MyInt="");
Set(MyVal="");
Set(XCoord="");
Set(YCoord="");
end proc:checkM:=proc()
global Mval;
local userM, userMval, userMConv;
userM:=Get(MInt::function):
userMval:=Get(MVal::realcons):
if intType=0 then
userMConv:=subs({x=u,y=v},userM);
else
userMConv:=subs({y=u,x=v},userM);
end if;
if userMConv=M and userMval=Mval then
Set(response="Correct!")
elif userMConv=M then
Set(response="The integral is correct, but the value is wrong. Try again.")
else
Set(response="The integral is wrong. Try again.")
end if;
end proc:hintM:=proc()
if intType=0 then
Set(MInt="Int(Int(rho,y=f(x)..g(x)),x=a..b)")
else
Set(MInt="Int(Int(rho,x=f(y)..g(y)),y=a..b)")
end if;
end proc:showM:=proc()
global Mval;
local MConv;
if intType=0 then
MConv:=subs({u=x,v=y},M);
else
MConv:=subs({u=y,v=x},M);
end if;
Set(MInt=MConv);
Set(MVal=Mval):
end proc:checkMy:=proc()
local userMy, userMyConv, userMyval:
global Myval:
userMy:=Get(MyInt::realcons):
userMyval:=Get(MyVal::realcons):
if intType=0 then
userMyConv:=subs({x=u,y=v},userMy);
else
userMyConv:=subs({x=v,y=u},userMy);
end if;
if userMyConv=My and userMyval=Myval then
Set(response="Correct!"):
elif userMyConv=My then
Set(response="The integral is correct, but the value is wrong. Try again."):
else
Set(response="The integral is wrong. Try again."):
end if:
end proc:hintMy := proc ()
if intType=0 then
Set(MyInt = "Int(Int(x*rho,y=f(x)..g(x)),x=a..b)")
else
Set(MyInt = "Int(Int(x*rho,x=f(y)..g(y)),y=a..b)")
end if;
end proc:showMy := proc ()
local MyConv;
if intType=0 then
MyConv:=subs({u=x,v=y},My);
else
MyConv:=subs({u=y,v=x},My);
end if;
Set(MyInt = MyConv);
Set(MyVal = value(My))
end proc:checkMx := proc ()
local userMx, userMxval, userMxConv;
global Mxval; userMx := Get(MxInt::realcons);
userMxval := Get(MxVal::realcons);
if intType=0 then
userMxConv:=subs({x=u,y=v},userMx);
else
userMxConv:=subs({x=v,y=u},userMx);
end if;
if userMxConv=Mx and userMxval=Mxval then
Set(response="Correct!")
elif userMxConv=Mx then
Set(response="The integral is correct, but the value is wrong. Try again.")
else
Set(response="The integral is wrong. Try again.")
end if;
end proc:hintMx := proc ()
if intType=0 then
Set(MxInt = "Int(Int(y*rho,y=f(x)..g(x)),x=a..b)")
else
Set(MxInt = "Int(Int(y*rho,x=f(y)..g(y)),y=a..b)")
end if;
end proc:showMx:=proc()
local MxConv;
if intType=0 then
MxConv:=subs({u=x,v=y},Mx);
else
MxConv:=subs({u=y,v=x},Mx);
end if;
Set(MxInt = MxConv);
Set(MxVal = value(Mx))
end proc:checkX:=proc()
local userX;
userX:=Get(XCoord::realcons);
if userX=X then
Set(response="Correct!")
else
Set(response="Incorrect. Try again.")
end if;
end proc:hintX:=proc ()
Set(XCoord="My/M")
end proc:showX:=proc ()
Set(XCoord=X)
end proc:checkY:=proc ()
local userY;
userY:=Get(YCoord::realcons);
if userY=Y then
Set(response="Correct!")
else
Set(response="Incorrect. Try again.")
end if;
end proc:hintY := proc ()
Set(YCoord = "Mx/M")
end proc:showY := proc ()
Set(YCoord = Y)
end proc:CenterOfMass:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN] (title="Center of Mass of a Plate with Variable Density",
[ valign=none, inset=0,
[ Button("New Problem", Evaluate(function = "ask"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ halign=none, valign=none,
[ halign=none, valign=none, inset=0,
TextBox[question](width=30, height=3, editable=false),
BoxColumn( border=true,
caption="Find the total mass, M. Enter the integral and value.",
[ halign=none,
"M = ",
TextBox[MInt](width = 30, height=2, background=turquoise),
" = ",
TextField[MVal](width=10, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintM"), background=orange),
Button("Check", Evaluate(function="checkM"), background=green),
Button("Show", Evaluate(function="showM"), background=yellow),
HorizontalGlue(),
HorizontalGlue()
]
)
],
[ border=true,
Plotter[graph](plot(0,x=0..6), width=300, height=200)
]
],
[ halign=none, valign=none,
[ border=true,
BoxColumn( border=true,
caption="Find the moment about the y-axis (integral and value):",
[ inset=0, spacing=0,
"My = ",
TextBox[MyInt](width=35, height=2, background=turquoise),
" = ",
TextField[MyVal](width=10, background=turquoise)
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintMy"), background=orange),
Button("Check", Evaluate(function="checkMy"), background=green),
Button("Show", Evaluate(function="showMy"), background=yellow),
HorizontalGlue()
]
),
BoxColumn( border=true,
caption="Find the x-coordinate of the center of mass:",
[ halign=none,
HorizontalGlue(),
"X = ",
TextField[XCoord](width=20, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintX"), background=orange),
Button("Check", Evaluate(function="checkX"), background=green),
Button("Show", Evaluate(function="showX"), background=yellow),
HorizontalGlue()
]
)
],
[ border=true,
BoxColumn( border=true,
caption="Find the moment about the x-axis (integral and value):",
[ inset=0, spacing=0,
"Mx = ",
TextBox[MxInt](width=35, height=2, background=turquoise),
" = ",
TextField[MxVal](width=10, background=turquoise)
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintMx"), background=orange),
Button("Check", Evaluate(function="checkMx"), background=green),
Button("Show", Evaluate(function="showMx"), background=yellow),
HorizontalGlue()
]
),
BoxColumn( border=true,
caption="Find the y-coordinate of the center of mass:",
[ halign=none,
HorizontalGlue(),
"Y = ",
TextField[YCoord](width=20, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintY"), background=orange),
Button("Check", Evaluate(function="checkY"), background=green),
Button("Show", Evaluate(function="showY"), background=yellow),
HorizontalGlue()
]
)
]
],
[ TextField[response](width=60)
],
[ "Programmer: Sarah Mannen",
HorizontalGlue(),
"copyright P. Yasskin 2008"
]
]
)
):Maplets[Display](CenterOfMass);