Final answer:
The volume of the solid obtained by rotating the region bound by y = 0 and y = x² - 2 about the x-axis is found by setting up an integral using the disk method. The limits of integration are the points where the parabola intersects the x-axis, which are -√2 and √2. The integral for the volume is ∫_-√2²√2 (x² - 2)² dx.
Step-by-step explanation:
To set up an integral that represents the volume of the solid obtained by rotating the region bounded by the curves y = 0 and y = x² - 2 about the x-axis, we use the disk method. The volume V can be expressed as:
V = ∫ᵽᵥ(x)² dx
Where ᵽᵥ(x) is the radius of the disk at a particular value of x, which is given by the equation y = x² - 2. The region of integration is from where the parabola intersects the x-axis. To find these intersection points, we set y to 0 and solve for x:
0 = x² - 2
x = √2, -√2
Thus, the limits of integration are from -√2 to √2. The integral that represents the volume is:
V = ∫_-√2²√2 (x² - 2)² dx