Step 1 :Given the piecewise function $f(x)=\left\{\begin{array}{l}x^{2}-4 x \text { for } x<-4 \\ 4 x-x^{2} \text { for } x \geq-4\end{array}\right.$, we are asked to find the integral of $f(x)$ from $-16$ to $4$.
Step 2 :To do this, we need to split the integral at $x=-4$, where the function changes.
Step 3 :We integrate $x^{2}-4 x$ from $-16$ to $-4$ and $4 x-x^{2}$ from $-4$ to $4$.
Step 4 :The integral of $x^{2}-4 x$ from $-16$ to $-4$ is 1824.
Step 5 :The integral of $4 x-x^{2}$ from $-4$ to $4$ is $-\frac{128}{3}$.
Step 6 :Adding these two results together, we find that the integral of $f(x)$ from $-16$ to $4$ is $\frac{5344}{3}$.
Step 7 :Thus, the final answer is \(\boxed{\frac{5344}{3}}\).