cardioide.mw

> with(plots):
 

> plot([1-sin(theta),theta,theta=0..2*Pi],coords=polar,thickness=2);
 

Plot_2d
 

> Aire := Int(Int(1*r, r = 0..1-sin(theta)), theta = 0..2*Pi);
Aire := Int((1-sin(theta))^2/2, theta = 0..2*Pi);
Aire := Int(1, theta = 0..2*Pi)/2-Int(sin(theta), theta = 0..2*Pi)+Int((sin(theta))^2, theta = 0..2*Pi)/2;
Aire := int(1, theta = 0..2*Pi)/2-int(sin(theta), theta = 0..2*Pi)+Int((1-cos(2*theta))/2, theta = 0..2*Pi)/2;
Aire := int(1, theta = 0..2*Pi)/2-int(sin(theta), theta = 0..2*Pi)+int((sin(theta))^2, theta = 0..2*Pi)/2;
 

 

 

 

 

Int(Int(r, r = 0 .. `+`(1, `-`(sin(theta)))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
Int(`+`(`*`(`/`(1, 2), `*`(`^`(`+`(1, `-`(sin(theta))), 2)))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
`+`(`*`(`/`(1, 2), `*`(Int(1, theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `-`(Int(sin(theta), theta = 0 .. `+`(`*`(2, `*`(Pi))))), `*`(`/`(1, 2), `*`(Int(`*`(`^`(sin(theta), 2)), theta = 0 .. `+`(`*`(2, `*`...
`+`(Pi, `*`(`/`(1, 2), `*`(Int(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(cos(`+`(`*`(2, `*`(theta)))))))), theta = 0 .. `+`(`*`(2, `*`(Pi)))))))
`+`(`*`(`/`(3, 2), `*`(Pi))) (1)
 

> x[G]:=C/Aire;
C:=(Int(Int((r*cos(theta))*r, r = 0..1-sin(theta)), theta = 0..2*Pi));
C:=Int(cos(theta)*(1-sin(theta))^3/3, theta = 0..2*Pi);
C:=int(cos(theta)*(1-sin(theta))^3/3, theta = 0..2*Pi);
x[G]:=(int(int((r*cos(theta))*r, r = 0..1-sin(theta)), theta = 0..2*Pi))/Aire;
 

 

 

 

 

`+`(`/`(`*`(`/`(2, 3), `*`(Int(`+`(`*`(`/`(1, 3), `*`(cos(theta), `*`(`^`(`+`(1, `-`(sin(theta))), 3))))), theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `*`(Pi)))
Int(Int(`*`(`^`(r, 2), `*`(cos(theta))), r = 0 .. `+`(1, `-`(sin(theta)))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
Int(`+`(`*`(`/`(1, 3), `*`(cos(theta), `*`(`^`(`+`(1, `-`(sin(theta))), 3))))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
0
0 (2)
 

> y[G]:=E/Aire;
E:=(Int(Int((r*sin(theta))*r, r = 0..1-sin(theta)), theta = 0..2*Pi));
E:=Int(sin(theta)*(1-sin(theta))^3/3, theta = 0..2*Pi);
E:=Int(sin(theta), theta = 0..2*Pi)/3-Int((sin(theta))^2, theta = 0..2*Pi)+Int((sin(theta))^3, theta = 0..2*Pi)-Int((sin(theta))^4, theta = 0..2*Pi)/3;
E:=Int(sin(theta), theta = 0..2*Pi)/3-Int((1-cos(2*theta))/2, theta = 0..2*Pi)+Int((sin(theta))*(1-(cos(theta))^2), theta = 0..2*Pi)-Int((1-cos(2*theta))^2/4, theta = 0..2*Pi)/3;
E:=int(sin(theta), theta = 0..2*Pi)/3-int((1-cos(2*theta))/2, theta = 0..2*Pi)+int((sin(theta))*(1-(cos(theta))^2), theta = 0..2*Pi)-Int(1-cos(2*theta)+(cos(2*theta))^2, theta = 0..2*Pi)/12;
E:=int(sin(theta), theta = 0..2*Pi)/3-int((1-cos(2*theta))/2, theta = 0..2*Pi)+int((sin(theta))*(1-(cos(theta))^2), theta = 0..2*Pi)-Int(1, theta = 0..2*Pi)/12+Int(cos(2*theta), theta = 0..2*Pi)/12-Int((cos(2*theta))^2, theta = 0..2*Pi)/12;
E:=int(sin(theta), theta = 0..2*Pi)/3-int((1-cos(2*theta))/2, theta = 0..2*Pi)+int((sin(theta))*(1-(cos(theta))^2), theta = 0..2*Pi)-int(1, theta = 0..2*Pi)/12+int(cos(2*theta), theta = 0..2*Pi)/12-Int((cos(4*theta)+1)/2, theta = 0..2*Pi)/12;
E:=int(sin(theta), theta = 0..2*Pi)/3-int((1-cos(2*theta))/2, theta = 0..2*Pi)+int((sin(theta))*(1-(cos(theta))^2), theta = 0..2*Pi)-int(1, theta = 0..2*Pi)/12+int(cos(2*theta), theta = 0..2*Pi)/12-int((cos(4*theta)+1)/2, theta = 0..2*Pi)/12;
y[G]:=value(%)/Aire;
 

 

 

 

 

 

 

 

 

 

-`/`(5, 6)
Int(Int(`*`(`^`(r, 2), `*`(sin(theta))), r = 0 .. `+`(1, `-`(sin(theta)))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
Int(`+`(`*`(`/`(1, 3), `*`(sin(theta), `*`(`^`(`+`(1, `-`(sin(theta))), 3))))), theta = 0 .. `+`(`*`(2, `*`(Pi))))
`+`(`*`(`/`(1, 3), `*`(Int(sin(theta), theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `-`(Int(`*`(`^`(sin(theta), 2)), theta = 0 .. `+`(`*`(2, `*`(Pi))))), Int(`*`(`^`(sin(theta), 3)), theta = 0 .. `+`(`*`(2, ...
`+`(`*`(`/`(1, 3), `*`(Int(sin(theta), theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `-`(Int(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(cos(`+`(`*`(2, `*`(theta)))))))), theta = 0 .. `+`(`*`(2, `*`(Pi))))), Int(`*...
`+`(`*`(`/`(1, 3), `*`(Int(sin(theta), theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `-`(Int(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(cos(`+`(`*`(2, `*`(theta)))))))), theta = 0 .. `+`(`*`(2, `*`(Pi))))), Int(`*...
`+`(`-`(Pi), `-`(`*`(`/`(1, 12), `*`(Int(`+`(1, `-`(cos(`+`(`*`(2, `*`(theta))))), `*`(`^`(cos(`+`(`*`(2, `*`(theta)))), 2))), theta = 0 .. `+`(`*`(2, `*`(Pi))))))))
`+`(`-`(Pi), `-`(`*`(`/`(1, 12), `*`(Int(1, theta = 0 .. `+`(`*`(2, `*`(Pi))))))), `*`(`/`(1, 12), `*`(Int(cos(`+`(`*`(2, `*`(theta)))), theta = 0 .. `+`(`*`(2, `*`(Pi)))))), `-`(`*`(`/`(1, 12), `*`(I...
`+`(`-`(`*`(`/`(7, 6), `*`(Pi))), `-`(`*`(`/`(1, 12), `*`(Int(`+`(`*`(`/`(1, 2), `*`(cos(`+`(`*`(4, `*`(theta)))))), `/`(1, 2)), theta = 0 .. `+`(`*`(2, `*`(Pi))))))))
`+`(`-`(`*`(`/`(5, 4), `*`(Pi))))
-`/`(5, 6) (3)
 

> p:=plot([1-sin(theta),theta,theta=0..2*Pi],coords=polar,thickness=2):
c:=pointplot([x[G],y[G]],symbolsize=20,symbol=solidcircle,color=red):
display(p,c);
 

Plot_2d
 

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