/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 17.10.1 ] */
/* [wxMaxima: title   start ]
Aplikace - procedury
   [wxMaxima: title   end   ] */


/* [wxMaxima: input   start ] */
Sm(f,g,a,b):=integrate(abs(f-g),x,a,b);    
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Sm(x^2,2-x,-2,1);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Sp(u,v,a,b):=integrate(v*abs(diff(u,t)),t,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
obsah:Sp(a*cos(t)^3,a*sin(t)^3,0,2*%pi);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(a>0,sin(t)>0)$ ev(obsah,nouns); forget(a>0,sin(t)>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
L(f,a,b):=integrate(sqrt(1+(diff(f,x))^2),x,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
L(log(cos(x)),0,a);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: answer  start ] */
positive;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: answer  start ] */
positive;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
delka:subst(log(tan(a)+sqrt(tan(a)^2+1)),asinh(tan(a)),%);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(cos(a)>0)$ delka2:trigsimp(delka); forget(cos(a)>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
radcan(delka2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Lp(u,v,a,b):=integrate(sqrt((diff(u,t))^2+(diff(v,t))^2),t,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(a>0)$ abs(4*Lp(a*cos(t)^3,a*sin(t)^3,0,%pi/2)); forget(a>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vx(f,a,b):=%pi*integrate(f^2,x,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vx((x^2/a)*sqrt(1-(x^2/a^2)),-a,a);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vy(f,a,b):=2*%pi*integrate(x*f,x,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vy(tan(x)/x,%pi/4,%pi/3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vpx(u,v,a,b):=%pi*integrate(v^2*abs(diff(u,t)),t,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
objem:Vpx(a*cos(t)^3,a*sin(t)^3,0,%pi);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(a>0,sin(t)>0)$ ev(objem,nouns); forget(a>0,sin(t)>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Vpy(u,v,a,b):=2*%pi*integrate(v*u*abs(diff(u,t)),t,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(a>0,t>0)$ 2*Vpy(a*((t^2-3)/(t^2+1)),a*((t*(t^2-3))/(t^2+1)),0,sqrt(3)); forget(a>0,t>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Qx(f,a,b):=2*%pi*integrate(f*sqrt(1+(diff(f,x))^2),x,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
plast:Qx(tan(x),0,%pi/4);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: answer  start ] */
positive;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
subst([asinh(1)=log(1+sqrt(1^2+1)),asinh(1/2)=log((1/2)+sqrt((1/2)^2+1))],plast);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
expand(%);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Qy(f,a,b):=2*%pi*integrate(x*sqrt(1+(diff(f,x))^2),x,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Qpx(u,v,a,b):=2*%pi*integrate(v*sqrt((diff(u,t))^2+(diff(v,t))^2),t,a,b);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
assume(a>0)$ 2*abs(Qpx(a*cos(t)^3,a*sin(t)^3,0,%pi/2)); forget(a>0)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
Qpy(u,v,a,b):=2*%pi*integrate(u*sqrt((diff(u,t))^2+(diff(v,t))^2),t,a,b);
/* [wxMaxima: input   end   ] */



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