To find the area under the curve \(y = \frac{4}{\sqrt{x}}\) from \(x = 1\) to \(x = 4\), we need to evaluate the definite integral:

\(\int_{1}^{4} 4x^{-0.5} \, dx\)

First, find the antiderivative of \(4x^{-0.5}\):

\(\int 4x^{-0.5} \, dx = \frac{4x^{0.5}}{0.5} = 8x^{0.5}\)

Now, evaluate this antiderivative from 1 to 4:

\(\left[ 8x^{0.5} \right]_{1}^{4} = 8 \sqrt{4} - 8 \sqrt{1}\)

\(= 8 \times 2 - 8 \times 1\)

\(= 16 - 8 = 8\)

Thus, the area of the region is 8.