Calculate $\int_{0}^{2 \pi} f(x) d x$, where \[ f(x)=\left\{\begin{aligned} \sin (x), & x \leq \pi \\ -4 \sin (x), & x>\pi \end{aligned}\right. \] (Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ \int_{0}^{2 \pi} f(x) d x= \]

Step 1 ：Define the function \(f(x)\) as follows: \(f(x)=\sin (x)\) for \(x \leq \pi\) and \(f(x)=-4 \sin (x)\) for \(x>\pi\).

Step 2 ：Calculate the integral of \(f(x)\) from 0 to \(\pi\), which is \(2\).

Step 3 ：Calculate the integral of \(f(x)\) from \(\pi\) to \(2\pi\), which is \(8\).

Step 4 ：Add the two integrals together to get the total integral from 0 to \(2\pi\), which is \(10\).

Step 5 ：Final Answer: The integral of the function from 0 to \(2\pi\) is \(\boxed{10}\).