Final answer:
The volume of the solid obtained by rotating the region bounded by y = x, y = 0, x = 0, and x = 2 about the x-axis is (8/3)π cubic units, calculated using the method of discs or washers by integrating the area of circular cross-sections from x = 0 to x = 2.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by y = x, y = 0, x = 0, and x = 2 about the x-axis, we will use the method of discs or washers in calculus. This involves integrating the area of the circular cross-sections perpendicular to the axis of rotation along the interval from x = 0 to x = 2.
The area A of a cross-section at a given x-value is πr2, where r is the distance from the x-axis to the curve y = x, which is simply y itself since the curve y = x is being rotated around the x-axis.
Therefore, the area A is πx2, and the volume V can be found by integrating this area over the limits of x from 0 to 2:
V = ∫02 πx2 dx = π∫02 x2 dx
This integral evaluates to πx3/3 from 0 to 2, which gives us: V = π(23/3) - π(03/3) = (8/3)π cubic units.