Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, x = 0, and x = 2 about the x-axis, we can use the disk method and integrate πx⁴ over the interval 0 to 2 with respect to x.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, x = 0, and x = 2 about the x-axis, we can use the disk method. The disk method involves integrating the cross-sectional areas of infinitesimally thin disks perpendicular to the axis of rotation.
The equation of the curve y = x² consists of positive values of x from 0 to 2, so we integrate with respect to x. The radius of each disk is given by the y-value of the curve, which is x². Therefore, the area of each disk is π(x²)² = πx⁴.
Integrating πx⁴ over the interval 0 to 2 gives us the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis: V = ∫(0 to 2) πx⁴ dx.