Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=6x^2, y=0, x=1, and x=2 about the y-axis using the method of cylindrical shells, divide the region into infinitely thin cylindrical shells and sum up their volumes.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=6x^2, y=0, x=1, and x=2 about the y-axis using the method of cylindrical shells, we can divide the region into infinitely thin cylindrical shells and sum up their volumes. Each shell has a thickness of dx and a radius of x. The height of each shell is given by the difference between the upper and lower y-values of the curves at x. Therefore, the volume of each shell is given by 2πx(6x^2)dx. We can integrate this expression from x=1 to x=2 to find the total volume:
V = ∫12 2πx(6x^2)dx
Simplifying the integral and evaluating it gives the volume of the solid.