Find the area of the region enclosed by the curves $y=7 \sin x$ and $y=\sin (7 x), 0 \leq x \leq \pi$.

Step 1 ：First, we need to find the points of intersection of the two curves $y=7 \sin x$ and $y=\sin (7 x)$ over the interval $0 \leq x \leq \pi$.

Step 2 ：The points of intersection are approximately $x = -2.85256594e-06, -1.63298958e-06, -2.28745010e-07, 0.00000000e+00, 3.14159265e+00, 3.14159288e+00, 3.14159429e+00, 3.14159551e+00$.

Step 3 ：Next, we need to integrate the absolute difference of the two functions over the given interval. This involves determining which function is greater over each subinterval defined by the points of intersection, and integrating the difference accordingly.

Step 4 ：After performing the integration, we find that the area between the curves $y=7 \sin x$ and $y=\sin (7 x)$ from $0$ to $\pi$ is approximately 13.71.

Step 5 ：So, the final answer is \(\boxed{13.71}\)