To find the volume of the solid generated when region R is rotated about the x-axis, we can use the disk method.
First, we need to find the points of intersection between the line y = 1 and the curve y = 1/4x^2. Setting the equations equal to each other:
1 = 1/4x^2
Multiplying both sides by 4x^2:
4x^2 = 1
Dividing both sides by 4:
x^2 = 1/4
Taking the square root of both sides:
x = ±1/2
So, the points of intersection are (1/2, 1) and (-1/2, 1).
Now, we integrate with respect to x from -1/2 to 1/2 to find the volume of the solid:
V = π ∫(-1/2)^(1/2) (1/4x^2)^2 dx
V = π ∫(-1/2)^(1/2) 1/16x^4 dx
V = π [(1/80)x^5]_(-1/2)^(1/2)
V = π ((1/80) ∙ (1/2)^5 - (1/80) ∙ (-1/2)^5)
V = π ((1/80) ∙ (1/32) - (1/80) ∙ (-1/32))
V = π ((1/2560) + (1/2560))
V = π ∙ (1/1280)
So, the volume of the solid generated when region R is rotated about the x-axis is π/1280.
