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workaround...

vv 13305
November 21 2024
3 0 
latex(``* e, 'output'='string')

Answer...

vv 13305
November 19 2024
0 7
> 

Int(1/( (a^2+x^2)^(3/2) * x ), x=0..infinity):
% = value(%) assuming a>0;

Int(1/((a^2+x^2)^(3/2)*x), x = 0 .. infinity) = infinity

 (1)

You probably want:

 

> 

Int(1/( (a^2+x^2)^(3/2) ) * x, x=0..infinity):
% = simplify(value(%)) assuming a>0;

Int(x/(a^2+x^2)^(3/2), x = 0 .. infinity) = 1/a

 (2)

Groebner...

vv 13305
November 15 2024
3 4

A smaller degree system but with 7 equations

 

> 

restart;

> #

with(RootFinding):
> 

b := [114.069^2, 109.2389^2, 103.892^2, 99.76348^2, 97.24296^2];
b := floor~(b);  # optional
f := [x1^2+x2^2+x3^2+2*(x1*x2*cos(x4)+x2*x3*cos(x5)+x1*x3*cos(x4+x5))-b[1],
      x1^2+x2^2+x3^2+2*(x1*x2*cos(2*x4)+x2*x3*cos(2*x5)+x1*x3*cos(2*(x4+x5)))-b[2],
      x1^2+x2^2+x3^2+2*(x1*x2*cos(3*x4)+x2*x3*cos(3*x5)+x1*x3*cos(3*(x4+x5)))-b[3],
      x1^2+x2^2+x3^2+2*(x1*x2*cos(4*x4)+x2*x3*cos(4*x5)+x1*x3*cos(4*(x4+x5)))-b[4],
      x1^2+x2^2+x3^2+2*(x1*x2*cos(5*x4)+x2*x3*cos(5*x5)+x1*x3*cos(5*(x4+x5)))-b[5] ]:

[13011.73676, 11933.13727, 10793.54766, 9952.751942, 9456.193270]

  

[13011, 11933, 10793, 9952, 9456]

 (1)
> 

F:=[ eval( expand(f), [cos(x4)=x4, cos(x5)=x5, sin(x4)=y4, sin(x5)=y5] )[],
    x4^2+y4^2-1, x5^2+y5^2-1];

[2*x1*x3*x4*x5-2*x1*x3*y4*y5+2*x1*x2*x4+2*x2*x3*x5+x1^2+x2^2+x3^2-13011, 8*x1*x3*x4^2*x5^2-8*x1*x3*x4*x5*y4*y5+4*x1*x2*x4^2-4*x1*x3*x4^2-4*x1*x3*x5^2+4*x2*x3*x5^2+x1^2-2*x1*x2+2*x1*x3+x2^2-2*x2*x3+x3^2-11933, 32*x1*x3*x4^3*x5^3-32*x1*x3*x4^2*x5^2*y4*y5-24*x1*x3*x4^3*x5+8*x1*x3*x4^2*y4*y5-24*x1*x3*x4*x5^3+8*x1*x3*x5^2*y4*y5+8*x1*x2*x4^3+8*x2*x3*x5^3+18*x1*x3*x4*x5-2*x1*x3*y4*y5-6*x1*x2*x4-6*x2*x3*x5+x1^2+x2^2+x3^2-10793, 128*x1*x3*x4^4*x5^4-128*x1*x3*x4^3*x5^3*y4*y5-128*x1*x3*x4^4*x5^2+64*x1*x3*x4^3*x5*y4*y5-128*x1*x3*x4^2*x5^4+64*x1*x3*x4*x5^3*y4*y5+16*x1*x2*x4^4+16*x1*x3*x4^4+128*x1*x3*x4^2*x5^2-32*x1*x3*x4*x5*y4*y5+16*x1*x3*x5^4+16*x2*x3*x5^4-16*x1*x2*x4^2-16*x1*x3*x4^2-16*x1*x3*x5^2-16*x2*x3*x5^2+x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2-9952, 512*x1*x3*x4^5*x5^5-512*x1*x3*x4^4*x5^4*y4*y5-640*x1*x3*x4^5*x5^3+384*x1*x3*x4^4*x5^2*y4*y5-640*x1*x3*x4^3*x5^5+384*x1*x3*x4^2*x5^4*y4*y5+160*x1*x3*x4^5*x5-32*x1*x3*x4^4*y4*y5+800*x1*x3*x4^3*x5^3-288*x1*x3*x4^2*x5^2*y4*y5+160*x1*x3*x4*x5^5-32*x1*x3*x5^4*y4*y5+32*x1*x2*x4^5+32*x2*x3*x5^5-200*x1*x3*x4^3*x5+24*x1*x3*x4^2*y4*y5-200*x1*x3*x4*x5^3+24*x1*x3*x5^2*y4*y5-40*x1*x2*x4^3-40*x2*x3*x5^3+50*x1*x3*x4*x5-2*x1*x3*y4*y5+10*x1*x2*x4+10*x2*x3*x5+x1^2+x2^2+x3^2-9456, x4^2+y4^2-1, x5^2+y5^2-1]

 (2)
> 

Groebner:-IsProper(F);          # ==> true i.e. the system is compatible

true

 (3)
> 

Groebner:-IsZeroDimensional(F); # --> true i.e. finite number of solutions

true

 (4)
> 

#Groebner:-Basis(F, tdeg(x1,x2,x3,x4,x5,y4,y5));
> 

Groebner:-SuggestVariableOrder(F);

y5, y4, x3, x2, x1, x5, x4

 (5)
> 

G:=Groebner:-Basis(F, tdeg(y5, y4, x3, x2, x1, x5, x4));

`[Length of output exceeds limit of 1000000]`

 (6)

Probably a plex basis is also doable but larger.

 

> 

nops(G)

292

 (7)

 

Solution...

vv 13305
November 13 2024
2 1

Probably "it's a shame" is not the best formulation.

> 

restart;
> 

gA:=x->mA*x+nA: gB:=x->mB*x+nB: gC:=x->mC*x+nC:
> 

Oa:= xa,ya : Ob:= xb,yb : Oc:= xc,yc :
> 

det:=LinearAlgebra:-Determinant:
> 

Numeric:=NULL:
> 

# numeric (optional)
xa:=1: xb:=4: xc:=8: ya:=0: yb:=0: yc:=0:
mA:=2: mB:=-3: mC:=-1: nA:=4: nB:=-3: nC:=10:
Numeric:=explicit:
> 

sys:=
'det'(<a_, gA(a_),1; b_, gB(b_),1; Oc,1>),
'det'(<b_, gB(b_),1; c_, gC(c_),1; Oa,1>),
'det'(<c_, gC(c_),1; a_, gA(a_),1; Ob,1>):
> 

sol:=solve({sys}, {a_,b_,c_}, Numeric);  # A = [a_,gA(a_)] etc

{a_ = -2, b_ = -1, c_ = 10}, {a_ = 64/151, b_ = -86/25, c_ = -103/8}

 (1)
> 

with(plots):
> 

display(
plot([gA(x), gB(x), gC(x)], x=-16..13, color=green),                                             #gA,gB,gC
pointplot([[Oa],[Ob],[Oc]], symbol=solidcircle,symbolsize=12, color=gold),                       #Oa,Ob,Oc
pointplot( eval([[a_, gA(a_)], [b_, gB(b_)], [c_, gC(c_)]], sol[2]), symbolsize=20, color=red),  #A,B,C
plot( eval( [gA(a_) + (gB(b_)-gA(a_))/(b_-a_)*(x-a_),                                            #AB
             gB(b_) + (gC(c_)-gB(b_))/(c_-b_)*(x-b_),                                            #BC
             gC(c_) + (gA(a_)-gC(c_))/(a_-c_)*(x-c_)],  sol[2]),                                 #AC
     x=-16..13, color=blue)
);

 
> 

 

> 

 

P.S. In the worksheet the code is better formatted.
Download ABC-vv.mw

Cyclic polygon...

vv 13305
November 11 2024
1 2

n must be >3
The polygon with maximal area is the cyclic one.
Denote L[i] the sides, here L[i] = i, and r the radius of the circle.
Let a[i] be the angle A[i]OA[i+1], where A[1], ..., A[n] are the vertices
and O is the center of the circumscribed circle. Then 

Sum(arcsin(L[i]/r/2, i=1..n) = Pi  ==> r

restart
R := n -> local i,r; fsolve(sum(arcsin(i/r/2), i=1..n) - Pi, r =n/2 .. 2*n):

Vector( 17, k -> 'R'(k+3)=R(k+3));


n:=7:  # Graphic example
r:=R(n):
b[0]:=0:
for i to n do a[i] := 2*arcsin(i/r/2); b[i]:=b[i-1]+a[i] od:
MaxArea:=r^2/2*add(sin(a[i]), i=1..n); 
plot([seq]([cos(b[i]),sin(b[i])], i=0..n), axes=none);

        MaxArea := 54.72494665

evala...

vv 13305
October 30 2024
1 0 
evala(%);
#           2

Hint for a student solution "by hand":
-  Use  (a+b)^3 = a^3 + b^3 + 3*a*b*(a + b)

-- Obtain a cubic equation satisfied by your number x and factor it as (x-2)(...)=0. 

Modified sectors...

vv 13305
October 30 2024
2 2

                          

intat...

vv 13305
October 10 2024
3 0

intat(f(x), x=a)  computes an antiderivative F(x) of f(x) and returns F(a).

Two antiderivatives may differ by an additive constant, depending e,g. on the method used to compute it.

 intat is useful when it is desired to subsequently perform a change of variable, or in the case of symbolic manipulation of the solutions of differential equations.

 

So, for example,

> 

intat(x^2, x=1);

1/3

 (1)

has not much sense, unless you know which antiderivative was computed.

 

> 

restart;
> 

f := sin(x)*cos(x) + 1/x;

sin(x)*cos(x)+1/x

 (2)
> 

F1 := intat(f, x=a);

ln(a)+(1/2)*sin(a)^2

 (3)
> 

F2 := intat(simplify(f), x=a);

ln(a)-(1/4)*cos(2*a)

 (4)
> 

simplify(F1-F2);

1/4

 (5)
> 

F3 := Int(f, x=0..a); value(%);

Int(sin(x)*cos(x)+1/x, x = 0 .. a)

  

infinity

 (6)

Note. When Intat(f(x), x=a)  is replaced by Int(f(x). x=0..a), odesolve will verify it, but 0 could be out of dom(f), so, nonsense!

 

Area only but fast...

vv 13305
October 07 2024
2 1 
1/2*evalf(Int(min(-6*cos(t), 2-2*cos(t))^2, t=Pi/2..3*Pi/2));
identify(%);
#                          15.70796327
#                              5 Pi

Using Edge is ok...

vv 13305
October 02 2024
1 0

Using Edge is ok, Firefox too; maybe try some font settings:

 

Pell type reduction...

vv 13305
September 30 2024
4 0

The answer for your previous problem works here too.

restart;
eq:=x^2+y^2-N*(1+x*y):  # [x=0, y=sqrt(N)] and [x=sqrt(N),y=0] are solutions!
XY:=[X=x, Y=2*y-x*N]: xy:=solve(XY,[x,y])[]:
EQ:=simplify(eval(eq,xy)):
for N in [9, 49 , 729] do
  SOL:=isolve(EQ):
  sol[N]:=map(u -> simplify(eval(xy, u)), [SOL]);
  num:={seq}(simplify(sol[N])[],_Z1=0..2);
  print('N'=N, select(type,num, [anything=integer,anything=integer]));
od:

N = 9, {[x = -1497363, y = -13307787], 
  [x = -1497363, y = -168480], [x = -168480, y = -1497363], 
  [x = -168480, y = -18957], [x = -18957, y = -168480], 
  [x = -18957, y = -2133], [x = -2133, y = -18957], 
  [x = -2133, y = -240], [x = -240, y = -2133], 
  [x = -240, y = -27], [x = -27, y = -240], [x = -27, y = -3], 
  [x = -3, y = -27], [x = -3, y = 0], [x = 0, y = -3], 
  [x = 0, y = 3], [x = 3, y = 0], [x = 3, y = 27], 
  [x = 27, y = 3], [x = 27, y = 240], [x = 240, y = 27], 
  [x = 240, y = 2133], [x = 2133, y = 240], 
  [x = 2133, y = 18957], [x = 18957, y = 2133], 
  [x = 18957, y = 168480], [x = 168480, y = 18957], 
  [x = 168480, y = 1497363], [x = 1497363, y = 168480], 
  [x = 1497363, y = 13307787]}

N = 49, {[x = -96687343207, y = -4735705783543], 
  [x = -96687343207, y = -1974033600], 
  [x = -1974033600, y = -96687343207], 
  [x = -1974033600, y = -40303193], 
  [x = -40303193, y = -1974033600], [x = -40303193, y = -822857], 
  [x = -822857, y = -40303193], [x = -822857, y = -16800], 
  [x = -16800, y = -822857], [x = -16800, y = -343], 
  [x = -343, y = -16800], [x = -343, y = -7], [x = -7, y = -343], 
  [x = -7, y = 0], [x = 0, y = -7], [x = 0, y = 7], 
  [x = 7, y = 0], [x = 7, y = 343], [x = 343, y = 7], 
  [x = 343, y = 16800], [x = 16800, y = 343], 
  [x = 16800, y = 822857], [x = 822857, y = 16800], 
  [x = 822857, y = 40303193], [x = 40303193, y = 822857], 
  [x = 40303193, y = 1974033600], [x = 1974033600, y = 40303193], 
  [x = 1974033600, y = 96687343207], 
  [x = 96687343207, y = 1974033600], 
  [x = 96687343207, y = 4735705783543]}

N = 729, {[x = -4052517025117644747, y = -2954279352292037818803], 
  [x = -4052517025117644747, y = -5559018725201760], 
  [x = -5559018725201760, y = -4052517025117644747], 
  [x = -5559018725201760, y = -7625554438293], 
  [x = -7625554438293, y = -5559018725201760], 
  [x = -7625554438293, y = -10460313837], 
  [x = -10460313837, y = -7625554438293], 
  [x = -10460313837, y = -14348880], 
  [x = -14348880, y = -10460313837], [x = -14348880, y = -19683], 
  [x = -19683, y = -14348880], [x = -19683, y = -27], 
  [x = -27, y = -19683], [x = -27, y = 0], [x = 0, y = -27], 
  [x = 0, y = 27], [x = 27, y = 0], [x = 27, y = 19683], 
  [x = 19683, y = 27], [x = 19683, y = 14348880], 
  [x = 14348880, y = 19683], [x = 14348880, y = 10460313837], 
  [x = 10460313837, y = 14348880], 
  [x = 10460313837, y = 7625554438293], 
  [x = 7625554438293, y = 10460313837], 
  [x = 7625554438293, y = 5559018725201760], 
  [x = 5559018725201760, y = 7625554438293], 
  [x = 5559018725201760, y = 4052517025117644747], 
  [x = 4052517025117644747, y = 5559018725201760], 
  [x = 4052517025117644747, y = 2954279352292037818803]}

With sol[N] you may inspect the general solution depending on _Z1.


 

Solution...

vv 13305
September 30 2024
5 1

 

isolve is old and should be updated.

Actually isolve knows to solve generalized Pell equations. It only needs a little help to convert the diophantine quadratic to generalized Pell form.

At the end we must filter the obtained numeric solutions in order to eliminate the non-integer ones due to the form of xy, see below.

 

> 

restart;
> 

eq:=x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3:
> 

XY:=[X = 2*x-12*y+4, Y = -5*y+3]; # via complete squares

[X = 2*x-12*y+4, Y = -5*y+3]

 (1)
> 

xy:=solve(XY,[x,y])[];

[x = (1/2)*X+8/5-(6/5)*Y, y = -(1/5)*Y+3/5]

 (2)
> 

EQ:=simplify(eval(eq,xy));

(1/4)*X^2+19/5-(6/5)*Y^2

 (3)
> 

SOL:=isolve( EQ ):
> 

sol:=map(u -> simplify(eval(xy, u)), [SOL]);

[[x = (1/10)*(30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1]]

 (4)
> 

nops(sol)

8

 (5)
> 

num:={seq}(simplify(sol)[],_Z1=0..20): # some numeric solutions
> 

numsols:=select(type,num,  [anything=integer,anything=integer]);

{[x = 3, y = 1], [x = 3, y = 3], [x = 5, y = 1], [x = 5, y = 7], [x = 29, y = 3], [x = 29, y = 53], [x = 75, y = 7], [x = 75, y = 141], [x = 603, y = 53], [x = 603, y = 1151], [x = 1613, y = 141], [x = 1613, y = 3083], [x = 13205, y = 1151], [x = 13205, y = 25257], [x = 35379, y = 3083], [x = 35379, y = 67673], [x = 289875, y = 25257], [x = 289875, y = 554491], [x = 776693, y = 67673], [x = 776693, y = 1485711], [x = 6364013, y = 554491], [x = 6364013, y = 12173533], [x = 17051835, y = 1485711], [x = 17051835, y = 32617957], [x = 139718379, y = 12173533], [x = 139718379, y = 267263223], [x = 374363645, y = 32617957], [x = 374363645, y = 716109331], [x = 3067440293, y = 267263223], [x = 3067440293, y = 5867617361], [x = 8218948323, y = 716109331], [x = 8218948323, y = 15721787313], [x = 67343968035, y = 5867617361], [x = 67343968035, y = 128820318707], [x = 180442499429, y = 15721787313], [x = 180442499429, y = 345163211543], [x = 1478499856445, y = 128820318707], [x = 1478499856445, y = 2828179394181], [x = 3961516039083, y = 345163211543], [x = 3961516039083, y = 7577868866621], [x = 32459652873723, y = 2828179394181], [x = 32459652873723, y = 62091126353263], [x = 86972910360365, y = 7577868866621], [x = 86972910360365, y = 166367951854107], [x = 712633863365429, y = 62091126353263], [x = 712633863365429, y = 1363176600377593], [x = 1909442511888915, y = 166367951854107], [x = 1909442511888915, y = 3652517071923721], [x = 15645485341165683, y = 1363176600377593], [x = 15645485341165683, y = 29927794081953771], [x = 41920762351195733, y = 3652517071923721], [x = 41920762351195733, y = 80189007630467743], [x = 343488043642279565, y = 29927794081953771], [x = 343488043642279565, y = 657048293202605357], [x = 920347329214417179, y = 80189007630467743], [x = 920347329214417179, y = 1760505650798366613], [x = 7541091474788984715, y = 657048293202605357], [x = 7541091474788984715, y = 14425134656375364071], [x = 20205720480365982173, y = 1760505650798366613], [x = 20205720480365982173, y = 38650935309933597731], [x = 165560524401715384133, y = 14425134656375364071], [x = 165560524401715384133, y = 316695914147055404193], [x = 443605503238837190595, y = 38650935309933597731], [x = 443605503238837190595, y = 848560071167740783457], [x = 3634790445362949466179, y = 316695914147055404193], [x = 3634790445362949466179, y = 6952884976578843528163], [x = 9739115350774052210885, y = 848560071167740783457], [x = 9739115350774052210885, y = 18629670630380363638311], [x = 79799829273583172871773, y = 6952884976578843528163], [x = 79799829273583172871773, y = 152646773570587502215381], [x = 213816932213790311448843, y = 18629670630380363638311], [x = 213816932213790311448843, y = 409004193797200259259373], [x = 1751961453573466853712795, y = 152646773570587502215381], [x = 1751961453573466853712795, y = 3351276133576346205210207], [x = 4694233393352612799663629, y = 409004193797200259259373], [x = 4694233393352612799663629, y = 8979462592908025340067883], [x = 38463352149342687608809685, y = 3351276133576346205210207], [x = 38463352149342687608809685, y = 73575428165109029012409161], [x = 103059317721543691281150963, y = 8979462592908025340067883], [x = 103059317721543691281150963, y = 197139172850179357222234041], [x = 844441785831965660540100243, y = 73575428165109029012409161], [x = 2262610756480608595385657525, y = 197139172850179357222234041]}

 (6)
> 

nops(numsols);

82

 (7)
> 

{seq}( eval(eq, s), s=numsols); #check

{0}

 (8)
> 

 

 

Download dioph-sols-vv.mw

andmap...

vv 13305
September 26 2024
3 0

 

select(u -> nops(u)=m and andmap(isprime,u), L);

 

DirectSearch...

vv 13305
September 24 2024
3 1

 

> 

restart;
> 

dis:= (A,B) -> sqrt( (A[1]-B[1])^2 + (A[2]-B[2])^2 ):
P:= unapply(dis(A,B)+dis(B,C)+dis(C,A), [A,B,C]):
> 

A0:=[0,0]:
A1:=[u1,0]:
A2:=[u2,u3]:
A3:=v1*~A1:
A4:=A1+v2*~(A2-A1):
A5:=A1+v3*~(A2-A1):
A6:=A0+v4*~(A2-A0):
A7:=A0+v5*~(A2-A0):
A8:=A6+v6*~(A4-A6):
> 

eps:=0.05:
> 

perim:=
P(A0,A3,A6)=10, P(A6,A3,A4)=15, P(A3,A1,A4)=11,
P(A6,A8,A7)=9,  P(A8,A5,A7)=13, P(A8,A4,A5)=12,
P(A7,A5,A2)=20:
> 

vars := u1=1..50, u2=1..50, u3=1..50,
v1=eps ..1-eps, v2=eps ..1-eps, v3=eps ..1-eps, v4=eps ..1-eps, v5=eps ..1-eps, v6=eps ..1-eps:
> 

inisol:=[u1 = 8, u2 = 3, u3 = 12, v1 = 0.5, v2 = 0.3, v3 = 0.5, v4 = 0.09, v5 = 0.3, v6 = 0.5]:
> 

sol:=DirectSearch:-SolveEquations([perim], [vars], initialpoint=inisol );

[7.928429359693041*10^(-17), Vector(7, {(1) = 0.1798504456e-8, (2) = -0.1393335225e-8, (3) = -0.7304805649e-9, (4) = 0.7992083439e-8, (5) = -0.1359651947e-8, (6) = 0.1542570516e-8, (7) = 0.2339469063e-8}), [u1 = 8.211758237046912, u2 = 1.8442345202064532, u3 = 12.033442054587029, v1 = .5526589984562774, v2 = .26313930832995064, v3 = .5414796127553622, v4 = 0.7943775626427665e-1, v5 = .2941739734925204, v6 = .485843681048224], 5405]

 (1)
> 

plots:-display(plot(eval([A0,A1,A2,A0],sol[3])), plot(eval([A3,A4,A6,A3],sol[3])), plot(eval([A5,A7,A8,A5],sol[3])), scaling=constrained);

 
> 

eval([perim], sol[3]);

[HFloat(10.000000001798504) = 10, HFloat(14.999999998606665) = 15, HFloat(10.99999999926952) = 11, HFloat(9.000000007992083) = 9, HFloat(12.999999998640348) = 13, HFloat(12.00000000154257) = 12, HFloat(20.00000000233947) = 20]

 (2)
> 

eval(seq("A"||i=A||i,i=0..8), sol[3])

"A0" = [0, 0], "A1" = [HFloat(8.211758237046912), 0], "A2" = [HFloat(1.8442345202064532), HFloat(12.033442054587029)], "A3" = [HFloat(4.538302082851432), 0], "A4" = [HFloat(6.536212450422957), HFloat(3.166471619072571)], "A5" = [HFloat(4.763873960641556), HFloat(6.515863543831874)], "A6" = [HFloat(0.14650185231032542), HFloat(0.9559096369525809)], "A7" = [HFloat(0.5425257968612042), HFloat(3.539925463989865)], "A8" = [HFloat(3.2509023701302153), HFloat(2.029897207531015)]

 (3)
> 

 


(Edited - a typo).

Download perims-vv.mw

infinite...

vv 13305
September 20 2024
3 1

The curve length is infinite. The integrand has a singularity at t=ln(Pi/2):

restart;
f:= t -> sqrt((sin(t/cos(exp(t))) + t*(1/cos(exp(t)) + t*exp(t)*sin(exp(t))/cos(exp(t))^2)*cos(t/cos(exp(t))))^2 + (cos(t) - t*sin(t))^2 + 2*t):
a:=ln(Pi/2): evalf(a);
#                          0.4515827054

Digits:=15:
:-int(f, a..a+1/10, numeric, epsilon=1e-5);
#                     3.33001574946940* 10^13
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