Final answer:
To find the volume of the solid created by rotating the region between y = 8x² and y = 0 about the x-axis, one must integrate the area of disks from the method of disks, using the formula πr², over the defined interval on the x-axis.
Step-by-step explanation:
The question asks us to find the volume of a solid obtained by rotating the region bounded by the curves y = 8x² and y = 0 about the x-axis. To find the volume of this solid, we can use the method of disks, which involves integrating the area of a sequence of circular disks along the x-axis from the starting point to the ending point of the region.
The area of one such disk, which is a circle, is πr², where r is the radius of the circle. Since we're rotating around the x-axis and the equation given is y = 8x², the radius r is 8x² and the area of one disk is π(8x²)². To find the total volume, we integrate this expression with respect to x over the interval of interest:
∫ π(8x²)² dx.
This integral will give us the volume of the solid after evaluating it over the specific x-interval for which the area is bounded.