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.)
\[
\int_{0}^{2 \pi} f(x) d x=
\]
Answer
Final Answer: The integral of the function from 0 to \(2\pi\) is \(\boxed{10}\).
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 \(\boxed{10}\).
