Final answer:
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the method of cylindrical shells. The volume can be calculated by evaluating the integral of the formula for the volume of each cylindrical shell. The resulting volume is π(36-6ln(6)-6).
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πrhΔx, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and Δx is the width of the shell. In this case, the distance from the x-axis to the curve y = x-1 is the radius of each shell, the height of each shell is the difference between the y-values of the curves y = x-1 and y = 0, and Δx is the infinitesimal width of each shell.
The volume of the solid is obtained by summing the volumes of all the cylindrical shells. We can do this by integrating the formula V = 2πrhΔx from x = 0 to x = 6. The integral becomes ∫[0,6] 2π(x-1)(x-0) dx. Evaluating this integral will give us the volume of the solid.
Simplifying the integral and evaluating it, we get V = π(36-6ln(6)-6). So, the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is π(36-6ln(6)-6).