2. $[2 / 5$ Points $]$ DETAILS PREVIOUS ANSWERS SCALC9 5.2.013.EP. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the solid obtained by rotating the region bounded by the given curves about the specified line. \[ y=\sqrt{x-1}, y=0, x=4 \text {; about the } x \text {-axis } \] Set up an integral that can be used to determine the volume $V$ of the solid. \[ V=\coprod^{4}[\square] d x \]

Step 1 ：Consider the solid obtained by rotating the region bounded by the given curves about the specified line. The curves are \(y=\sqrt{x-1}\), \(y=0\), and \(x=4\); and the rotation is about the x-axis.

Step 2 ：Set up an integral that can be used to determine the volume \(V\) of the solid. The volume of such a solid can be found using the disk method, which involves integrating the area of cross-sectional disks.

Step 3 ：The area of each disk is given by \(\pi r^2\), where \(r\) is the distance from the disk to the axis of rotation. In this case, the radius is given by the function \(y=\sqrt{x-1}\), so the area of each disk is \(\pi (\sqrt{x-1})^2 = \pi (x-1)\).

Step 4 ：The volume is then found by integrating this area from the leftmost to the rightmost x-values of the region, which are \(x=1\) and \(x=4\) respectively. Therefore, the integral that gives the volume of the solid is \(V = \int_{1}^{4} \pi (x-1) dx\).

Step 5 ：Finally, the integral can be calculated to find the volume of the solid. The final answer is \(V = \boxed{\frac{9\pi}{2}}\).