The volume \(V\) of the solid of revolution about the \(x\)-axis is given by the integral:

\(V = \pi \int_{1}^{4} y^2 \, dx\)

Substitute \(y = 3x^{\frac{1}{4}}\) into the integral:

\(V = \pi \int_{1}^{4} (3x^{\frac{1}{4}})^2 \, dx = \pi \int_{1}^{4} 9x^{\frac{1}{2}} \, dx\)

Integrate \(9x^{\frac{1}{2}}\):

\(\int 9x^{\frac{1}{2}} \, dx = 9 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = 6x^{\frac{3}{2}}\)

Evaluate from 1 to 4:

\(V = \pi \left[ 6x^{\frac{3}{2}} \right]_{1}^{4} = \pi \left( 6 \cdot 4^{\frac{3}{2}} - 6 \cdot 1^{\frac{3}{2}} \right)\)

Calculate \(4^{\frac{3}{2}} = 8\):

\(V = \pi (6 \cdot 8 - 6 \cdot 1) = \pi (48 - 6) = 42\pi\)