To find the volume of the solid of revolution generated by revolving the planar region bounded by the parabola y = 2x^2 - 3, the x-axis, and the lines x = 0 and x = 1 about the y-axis, we can use the method of cylindrical shells.

Step 1: Sketching the region
First, let's sketch the given region in the xy-plane. The parabola y = 2x^2 - 3 is symmetric with respect to the y-axis and opens upwards. It intersects the x-axis at x = ±√(3/2). The region is bounded by the parabola, the x-axis, and the vertical lines x = 0 and x = 1.

Step 2: Determining the height of the cylindrical shells
The height of each cylindrical shell is given by the difference in the y-values of the parabola at the corresponding x-values. The equation of the parabola is y = 2x^2 - 3.

At x = 0, the height of the shell is y = 2(0)^2 - 3 = -3.
At x = 1, the height of the shell is y = 2(1)^2 - 3 = -1.

So, the height of each cylindrical shell varies from -3 to -1.

Step 3: Determining the radius of the cylindrical shells
The radius of each cylindrical shell is given by the corresponding x-value. Since the shells are being revolved about the y-axis, the radius is the distance from the y-axis to the parabola at each x-value.

At x = 0, the radius of the shell is 0.
At x = 1, the radius of the shell is 1.

So, the radius of each cylindrical shell varies from 0 to 1.

Step 4: Calculating the volume of each cylindrical shell
The volume of each cylindrical shell is given by the formula V = 2πrh, where r is the radius and h is the height.

For this problem, the radius and height of each shell vary depending on the x-value. So, we need to integrate the volume of each shell over the interval [0, 1].

The integral for the volume of each cylindrical shell is:

V = ∫[0,1] 2πx(2x^2 - 3) dx

Simplifying the integral:

V = 2π ∫[0,1] (2x^3 - 3x) dx
= 2π [∫(2x^3) dx - ∫(3x) dx]
= 2π [x^4 - (3/2)x^2] evaluated from x = 0 to x = 1
= 2π [(1^4 - (3/2)(1^2)) - (0^4 - (3/2)(0^2))]
= 2π [(1 - (3/2)) - (0 - 0)]
= 2π [(1/2)]
= π

Step 5: Final answer
The volume of the solid of revolution generated is π cubic units. Since the question asks for the