Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=sqrt(x-1), y=0, x=2, and x=6 about the x-axis, the method of cylindrical shells can be used. The volume can be calculated by integrating the product of the height, circumference, and thickness of each shell.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=sqrt(x-1), y=0, x=2, and x=6 about the x-axis, we can use the method of cylindrical shells. Each shell will have a height of y=sqrt(x-1), a circumference of 2πx, and a thickness of dx. Therefore, the volume can be calculated by integrating the product of these values from x=2 to x=6.
The integral expression for the volume is V = ∫[2,6] 2πx(sqrt(x-1)) dx. By evaluating this integral, we can find the exact value of the volume.
By using the appropriate formula and performing the integral, the volume of the solid can be calculated as follows:
V = ∫[2,6] 2πx(sqrt(x-1)) dx = 44π/15