Mathematics

The key is recognizing that you're using the washer method for volumes of revolution: when you rotate a region around the $x$-axis, the volume comes from stacking circular washers (rings with holes) perpendicular to the axis. Each washer at position $x$ has outer radius $R(x) = 3$ and inner radius $r(x) = kx$, so the volume integral is $V = \pi \int_0^2 [3^2 - (kx)^2] \, dx$. Set this equal to $12\pi$ and solve for $k$.

The key is recognizing that you're using the washer method for volumes of revolution: when you rotate a region around the $x$-axis, the volume comes from stacking circular washers (rings with holes) perpendicular to the axis. Each washer at position $x$ has outer radius $R(x) = 3$ and inner radius $r(x) = kx$, so the volume integral is $V = \pi \int_0^2 [3^2 - (kx)^2] \, dx$. Set this equal to $12\pi$ and solve for $k$.