Animation created with the following SageMath code:

f(x)=1/x
C = plot(f,.4,7,color='black', xmin=0, figsize=6.5)
A2 = plot(f,1,3,fill=True,color='black',fillcolor=hue(.3),figsize=6.5)
o = point((0,0), size=.00001)
B = C+A2+o

def A(s,t,a,b,fillcolor=hue(.6),fillalpha=.5):
    f(x) = s/(t*x)
    Cst = plot(f,.4/t,7/t,color='blue', xmin=0,ymax=2.5)
    Ast = plot(f,a/t,b/t,fill=True,color='blue',fillcolor=fillcolor,fillalpha=fillalpha,xmin=0,ymax=2.5)
    T = text("f(x) = {:.3f}/({:.3f} x)".format(float(s),float(t)),(2,2), fontsize='large')
    T += text("A(1,6) = A(1,3) + A(3,6)", (4,1.2), fontsize='x-large')
    if t == 3 and s == 3:
        T += text("A(1,3) + A(1,2)", (4.665,.9), fontsize='x-large')
    return Cst + Ast + T

areas1 = [B+A(1,t,3,6)+line([(3,1/3),(3/t,1/3)],alpha=.4,color='black')+line([(6,1/6),(6/t,1/6)],alpha=.4,color='black') for t in list(sxrange(1,3,.162841))+[3]]
areas2 = [B+A(s,3,3,6) for s in list(sxrange(1,3,.162841))+[3]]
M = animate(areas1+areas2+10*[areas2[-1]])
M.save('log.gif')

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