Final answer:
To find the volume of a solid obtained by rotating the given curve around the x-axis, one must set up an integral of πr², with r as the function y = 6/7x², and then integrate across the defined bounds of x.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curve y = 6/7x² about the x-axis, you can use the method of disks (also known as circular slices). The formula for the volume of each disk is V = πr²h, where r is the radius of a disk, and h is the thickness of a disk, which in the context of integrals translates to dx (an infinitesimally small slice of x).
The radius of each disk in this case depends on the y-value from the function, which is 6/7x². Therefore, the volume of the solid is found by integrating πr² as a function of x, which would be π(6/7x²)² from the specified limits of x.
So, the integral that you need to solve is:
∫ π(6/7x²)² dx
This integral will give the volume of the entire solid once evaluated from the lower to the upper bound of x. However, note that the limits of integration have not been specified in the problem, so you'll need to determine or be given these limits to find the actual numeric value of the volume.