Use the shell method to find the volume of the solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cone of radius 8 and height 12 . Model the situation on a set of axes by placing the center of the base of the cone at the origin and the cone's axis along the positive \( y \)-axis. Set up the integral that gives the volume of the solid using the shell method. Use nonnegative and increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. B. The volume is . Type an exact answer.)

To set up the integral using the shell method, we first need to understand the geometry of the cone and the cylindrical hole being drilled. The cone's radius decreases linearly from 8 at the base (where \( y = 0 \)) to 0 at the tip (where \( y = 12 \)). The relationship can be given as \( r(y) = \frac{2}{3}(12 - y) \). The cylindrical hole of radius 2 will affect the outer volume. The volume of the solid can be determined through cylindrical shells, where the height of each shell is calculated based on the cone's slope. The volume \( V \) is given by the integral: \[ V = 2\pi \int_{0}^{12} (r(y) - 2)(\text{height of shell}) \, dy \] Where the height of the shell is the radius of the cone as a function of y, \( \frac{2}{3}(12 - y) \) minus the constant hole radius 2. Thus, the integral becomes: \[ V = 2\pi \int_{0}^{12} \left( \frac{2}{3}(12 - y) - 2 \right) \, dy \] After simplifying, this sets up the framework to find the exact volume. Now, you can proceed to evaluate this integral to find the volume of the solid formed by the drilled cone.