Answer
$12 \pi$
Work Step by Step
The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$
Now, $V=2\pi \int_{0}^{1} 3x dx +2 \pi \int_1^4 (x) \sqrt {4x^{-1}-1} \ dx \\ = 2\pi \int_0^1 3x dx + 2 \pi \int_1^4 (4-x) \ dx \\=2\pi [\dfrac{3x^2}{2}]_0^1+2 \pi[4x-\dfrac{x^2}{2}]_1^4 \\ = 2\pi \times \dfrac{3}{2} +2 \pi (8-\dfrac{7}{2}) \\ =12 \pi$
