It doesn't have to be that complicated. You can find the area using two integrals. We know the lower parabolas intersect the upper parabola at and the lower parabolas intersect each other at Thus, the area of the middle region is simply

And due to symmetry, we have

Your approach should work, but I'm not sure how you determined the area "from one of the x-intercepts to the point that the two parabolas touch" to be 1.125.