Suppose that $f$ is a continuous function such that \[ \int_{1}^{2} f(x) d x=4 \] Determine the value of the definite integral \[ \int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} d x \]

Step 1 ：Let's use the substitution method to solve the integral. We can let \(u = \sqrt{x}\), then \(du = \frac{1}{2\sqrt{x}} dx\). This changes the limits of integration from 1 to 2, and the integral becomes \(\int_{1}^{2} 2f(u) du\).

Step 2 ：We know that \(\int_{1}^{2} f(x) dx = 4\), we can substitute this into the integral to find the value.

Step 3 ：So, the value of the definite integral \(\int_{1}^{4} \frac{f(\sqrt{x})}{\sqrt{x}} dx\) is \(\boxed{8}\).