I took the integral from 1 to 0 to get
∫1 0 - (3(x3-x)) dx
As you can see I only evaluated the integral from 0 to 1 because part of the curve is above the x axis and the other part is below the x axis. I will end up doubling the answer to get the area since it is symmetric.
For the integral -3(1/4 x4 - 1/2 x2)
This will give you an answer of -3[(1/4 - 1/2) - 0]
This comes out to -3(-.25)-0 which is .75. Now I double it and the answer is 1.50. That should be the area.