\pgfmathparse{8} \let\xr\pgfmathresult;
\pgfmathparse{4.5} \let\yt\pgfmathresult;
\pgfmathparse{-0.1} \let\xl\pgfmathresult;
\pgfmathparse{-0.1} \let\yb\pgfmathresult;


\clip (\xl,\yb) rectangle (\xr,\yt);
\draw [>=stealth,->] (\xl,0) -- (\xr,0) node [right] {$x$};
\draw [>=stealth,->] (0,\yb) -- (0,\yt) node [above] {$y$};

\pgfmathparse{1} \let\a\pgfmathresult;
\pgfmathparse{3.5} \let\b\pgfmathresult;
\pgfmathparse{1} \let\Alpha\pgfmathresult;
\pgfmathparse{4} \let\Beta\pgfmathresult;

\pgfmathparse{0.2} \let\YB\pgfmathresult;
\pgfmathparse{8} \let\YT\pgfmathresult;

\draw [color=blue,name path=a,samples=200,domain=\YB:\YT] plot ({\a/\x}, {\x});
\draw [color=blue,name path=b,samples=200,domain=\YB:\YT] plot ({\b/\x}, {\x});

\draw [color=orange,name path=alpha,samples=200,domain=\YB:\YT] plot ({\Alpha/(\x*\x*\x)}, {\x});
\draw [color=orange,name path=beta, samples=200,domain=\YB:\YT] plot ({\Beta/(\x*\x*\x)},  {\x});

\tikzfillbetween[of=a and b]{blue, opacity=0.1};
\tikzfillbetween[of=alpha and beta]{blue, opacity=0.1};

\foreach \t in {\a,\b} {
  \pgfmathparse{sqrt(\Alpha/\t)} \let\y\pgfmathresult;
  \pgfmathparse{sqrt(\t*\t*\t/\Alpha)}\let\x\pgfmathresult;
  \filldraw (\x,\y) circle(1pt);
}
\foreach \t in {\a,\b} {
  \pgfmathparse{sqrt(\Beta/\t)} \let\y\pgfmathresult;
  \pgfmathparse{sqrt(\t*\t*\t/\Beta)}\let\x\pgfmathresult;
  \filldraw (\x,\y) circle(1pt);
}


%Calculate area
\pgfmathparse{\b-\a} \let\A\pgfmathresult;
\pgfmathparse{\A*ln(\Beta/\Alpha)} \let\A\pgfmathresult;
\pgfmathparse{0.5*\A} \let\A\pgfmathresult;

\coordinate (P1) at (4,3);
\coordinate (P2) at (4+\A,4);

\draw [draw=black,fill=blue, opacity=0.1] (P1) rectangle (P2);
\draw [draw=black,fill=blue, opacity=0.1] (P1) rectangle (P2);
\draw [draw=black] (P1) rectangle (P2) node[color=black,pos=.5] {A};

