The graph is an inverted U parabola with x intercepts at -1 and 1. The vertex is at (0,1).
We can picture a square lamina of thickness dy and side 2x, assuming that the lamina extends to x on one side of the y axis and to -x on the other side, so that the y axis bisects the sides of the squares perpendicular to it. The volume of the lamina is 4x^2dy and the area of the solid S will be the sum of the volumes of these laminae. In the limit as dy approaches zero the sum will be the integral of 4x^2dy between the limits y=0 to 1. The vertex of the parabola is at y=1, which is the upper limit where the volume of the lamina will have shrunk to zero. Write the integral as S[0,1](4x^2dy)=4S[0,1]((1-y)dy)=4(y-y^2/2)[0,1]=4(1-1/2)=2, where S (for sum) represents integral, not the solid S, and we have the volume of the solid S=2.