Final answer:
To calculate the volume of the solid produced by revolving the region bounded by the provided equations around the x-axis, one must set the equations for the parabolas equal, find the intersection points within the given bounds, and use the method of disks or washers to integrate the area of the cross-sectional washers from the lower to the upper bound.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the given equations about the x-axis, we use the method of disks or washers. The equations represent two parabolas that open in opposite directions and two vertical lines that bound the region. To find the intersection points, we set the parabolic equations equal to each other and solve for x.
y = x² + 2 and y = −x² + 2x + 6 can be combined into 0 = 2x² − 2x − 4, which simplifies to x² - x - 2 = 0. Factoring, we get (x-2)(x+1) = 0, so the intersection points are x = 2 and x = -1. However, since we are given the bounds x = 0 and x = 3, we only consider the intersection in this interval, which is x = 2.
The volume V of the solid is calculated using the integral of the cross-sectional area A(x) from the lower bound to the upper bound:
V = ∫₃₀ A(x) dx
Here, A(x) represents the area of a representative washer:
A(x) = π[(−x² + 2x + 6)² - (x² + 2)²]
Integrating A(x) from x = 0 to x = 3 will give us the required volume.