The area between two curves can be computed symbolically in a manner which makes it clear what is the basic idea behind the problem.

From the `ContourPlot` one can see that we could calculate `x` as a function of `y` for the both curves, we have: 

    Solve[y^3 - x^3 + 3 x^2 - 6 x - 4 y + 4 == 0, {y, x}, Reals] // Quiet
    Solve[x^3 + y^3 + x - 2 y - 1 == 0, {y, x}, Reals] // Quiet
>     {{x -> Root[-4 + 4 y - y^3 + 6 #1 - 3 #1^2 + #1^3 &, 1]}}
      {{x -> Root[-1 - 2 y + y^3 + #1 + #1^3 &, 1]}}

Next we find the intersection points:

    {r1, r2, r3} = y /. Solve[ Root[-4 + 4 y - y^3 + 6 #1 - 3 #1^2 + #1^3 &, 1]
                               == Root[-1 - 2 y + y^3 + #1 + #1^3 &, 1], y]
>     { Root[67 - 382 #1 - 324 #1^2 - 95 #1^3 + 270 #1^4 + 216 #1^5 - 
              51 #1^6 - 72 #1^7 + 8 #1^9 &, 1], 
       Root[67 - 382 #1 - 324 #1^2 - 95 #1^3 + 270 #1^4 + 216 #1^5 - 
            51 #1^6 - 72 #1^7 + 8 #1^9 &, 2], 
       Root[67 - 382 #1 - 324 #1^2 - 95 #1^3 + 270 #1^4 + 216 #1^5 - 
            51 #1^6 - 72 #1^7 + 8 #1^9 &, 3]}

And here we have a symbolic solution:

    Integrate[
      Abs[Root[-4 + 4 y - y^3 + 6 #1 - 3 #1^2 + #1^3 &, 1] - 
          Root[-1 - 2 y + y^3 + #1 + #1^3 &, 1]],
      {y, Root[67 - 382 #1 - 324 #1^2 - 95 #1^3 + 270 #1^4 + 216 #1^5 - 
               51 #1^6 - 72 #1^7 + 8 #1^9 &, 1], 
          Root[67 - 382 #1 - 324 #1^2 - 95 #1^3 + 270 #1^4 + 216 #1^5 - 
               51 #1^6 - 72 #1^7 + 8 #1^9 &, 3]}]

Since there we have roots of nineth order polynomial and the third order polynomila roots are quite involved the system cannot integrate the expression exactly. Nontheless we can find a good estimate:

    NIntegrate[ Abs[Root[-4 + 4 y - y^3 + 6 #1 - 3 #1^2 + #1^3 &, 1] - 
                    Root[-1 - 2 y + y^3 + #1 + #1^3 &, 1]], {y, r1, r3}]
>      3.01605

    Rotate [Plot[{ConditionalExpression[-Root[-4 + 4 y - y^3 + 6 #1 - 
        3 #1^2 + #1^3 &, 1], r1 <= y <= r3], 
                 ConditionalExpression[-Root[-1 - 2 y + y^3 + #1 + #1^3 &, 1], 
                                        r1 <= y <= r3]},
              {y, -1.85, 1.55}, AxesLabel -> Automatic, 
              PlotRange -> {-1.85, 1.55}, AspectRatio -> Automatic, 
              Filling -> {1 -> {2}}], 90 Degree]
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/F9ZUA.gif