Final answer:
To find the volume of the solid of rotation formed by the given region, the disk method is used to integrate the area of circular disks from y=0 to y=9. The radius of each disk is expressed as a function of y and integrated to find the volume.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region enclosed by x=0, x=1, y=0, and y=9x^5 about the y-axis, we use the disk method. This involves integrating the area of circular disks, as the region is rotated around the y-axis, from the lower bound of y (which is 0) to the upper bound (which is 9).
The radius of each disk is the x-value of the function, which is given implicitly by the equation of y=9x^5. To express this as x as a function of y, we solve for x to get x=(y/9)^(1/5). The volume of a thin disc at a given y-value is V = πr^2h, where r is the radius of the disc, and h is its height (or thickness).
The formula for the volume of the solid is therefore an integral from 0 to 9 of π times the square of the radius x=(y/9)^(1/5) with respect to the y-axis:
V = ∫_0^9 π[(y/9)^(1/5)]^2 dy
Then, after integrating, we multiply by the thickness dy to get the overall volume of the rotating body.