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

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

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]