posted by bob .
Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
y = 6 x^6 , y = 6 x , x >= 0
Find the volume V of this solid.
The curves intersect at x = 0 and x = 1. The region bounded between those curves has y-separation of 6(x-x^6).
For the total enclosed area, integrate that function times dx from x=0 to x=1.
I forgot that you wanted the volume of the solid obtained by rotating the curves about the x axis. This changes the formula to
Integral of 36 pi (x^2 - x^12) dx
...0 to 1