asympt02

\begin{asy}
// file fodsp.asy
//
// to get fodsp.pdf, run `asy -f pdf fodsp.asy`
//
size(8cm);
import graph;
import fontsize; defaultpen(fontsize(9pt));
texpreamble("\usepackage{lmodern}");

pen[] fillpen={
  red, orange, yellow, green, lightblue, blue, darkblue
};

real
xmin=0, xmax=20,
ymin=0, ymax=20;

xaxis(xmin,xmax,RightTicks(Step=10,step=5));
yaxis(ymin,ymax, LeftTicks(Step=10,step=5));

real ra(real t, real a){return 3*a*sin(t)*cos(t)/(sin(t)^3+cos(t)^3);};
real r(real);
guide loop;

real a, a0=10, da=1;
int n=fillpen.length;

real t; pair p;
a=a0;
for(int i=0;i<n;++i){
  r=new real(real t){return ra(t,a);};
  loop =polargraph(r,0,pi/2)--cycle;
  filldraw(loop, 0.7fillpen[i]+0.3white,fillpen[i]);
  t=atan(2^(1/3));
  p=r(t)*(cos(t),sin(t));
  unfill(circle(p,0.7));
  label("$"+string(a)+"$",p);
  a-=da;
}

label("$r(\theta)=\displaystyle"
+"\frac{3 a \sin\theta\cos\theta}{\sin^3\theta+\cos^3\theta}$, "
+"$\theta=[0,\frac\pi2]$, "
+"$a="+string(a0-(n-1)*da)+"$--$"+string(a0)+"$"
,((xmin+xmax)/2,ymax),S);

shipout(bbox(paleyellow,Fill));
\end{asy}