To find the volume of the solid generated by revolving the region enclosed by the graph of y = x^2, the line x = 2, and the x-axis about the y-axis, you need to use the disk method.
First, let's find the limits of integration. The intersection points of y = x^2 and x = 2 are found by substituting x = 2 into y = x^2:
y = 2^2 = 4
So, the region of integration is y = 0 to y = 4.
To use the disk method, we need to express the radius (r) in terms of y. Since the region is being revolved about the y-axis, the radius is the distance from the axis of revolution to the curve at a given value of y.
The equation y = x^2 can be rewritten as x = √y. The distance from the y-axis to √y is √y. Therefore, the radius r is equal to √y.
The differential thickness of the disk is dy.
The volume of each individual disk is given by dV = π * r^2 * dy.
Substituting r = √y and integrating from y = 0 to y = 4, we get:
V = ∫[0 to 4] π * (√y)^2 * dy
V = ∫[0 to 4] π * y * dy
V = π * [y^2/2] [0 to 4]
V = π * (4^2/2 - 0^2/2)
V = π * (16/2)
V = 8π
Therefore, the volume of the solid generated by revolving the region enclosed by the graph of y = x^2, the line x = 2, and the x-axis about the y-axis is 8π.
So, the correct answer is A. 8π.