First, in the velocity equation you have a -3t^2 but when you integrate you have a positive.

But your expression below is correct:

81/4-108/4+36/4=(117-108)/4=9/4.

I don't think you have done anything wrong. Also, if you want to find the distance you need to take into account when the velocity is negative. If you look carefully (i.e. differentiate) the velocity expression, you will see that is positive on some intervals but negative on others. When you integrate to find the speed you need to consider this.