Find the volume V of the described solid S.The base of S is the region enclosed by the parabola y = 4 − 2x2 and the x−axis. Cross-sections perpendicular to the y−axis are squares.

To find the volume of the solid, we need to integrate the area of the cross-sections from the lowest to the highest y-value on the base.

The base of the solid is the region enclosed by the parabola y = 4 - 2x^2 and the x-axis. This parabola intersects the x-axis when y = 0, so we solve for x:

0 = 4 - 2x^2 2x^2 = 4 x^2 = 2 x = sqrt(2) or x = -sqrt(2)

So the base of the solid extends from y = 4 - 2(-sqrt(2))^2 = 0 to y = 4 - 2(sqrt(2))^2 = 0.

The cross-sections perpendicular to the y-axis are squares. The side length of each square is the distance across the base at that y-value, which is 2x. We can solve for x in terms of y:

y = 4 - 2x^2 2x^2 = 4 - y x^2 = 2 - y/2 x = sqrt(2 - y/2)

So the side length of each square is 2x = 2sqrt(2 - y/2), and the area of each square is (2sqrt(2 - y/2))^2 = 8(2 - y/2) = 16 - 4y.

Finally, we integrate this area from y = 0 to y = 4 to find the volume of the solid:

V = ∫ from 0 to 4 of (16 - 4y) dy = [16y - 2y^2] from 0 to 4 = (164 - 24^2) - (160 - 20^2) = 64 - 32 = 32 cubic units.