To solve this problem, we need to find the region enclosed between the two curves and sketch it.

The two curves are given as:

1. $y = 5 \cos(5x)$
2. $y = 5 - 5 \cos(5x)$

Step-by-step approach:

1. Determine the points of intersection: To find the enclosed region, we first need to determine where the two curves intersect. To do this, we set the two equations equal to each other:

   $5 \cos(5x) = 5 - 5 \cos(5x)$

   Simplifying this:

   $2 \cdot 5 \cos(5x) = 5$

   $\cos(5x) = \frac{1}{2}$

   The general solution for $\cos(5x) = \frac{1}{2}$ is:

   5x = \pm \frac{\pi}{3} + 2n\pi \quad \text{(where n is any integer)}

   Solving for $x$:

   $x = \pm \frac{\pi}{15} + \frac{2n\pi}{5}$

   Considering the range $0 \leq x \leq \frac{\pi}{5}$, the solutions for $x$ are:

   $x = \frac{\pi}{15}$

2. Sketch the region: We now know that the curves intersect at $x = \frac{\pi}{15}$. To sketch the region between the curves, plot the curves $y = 5 \cos(5x)$ and $y = 5 - 5 \cos(5x)$ from $x = 0$ to $x = \frac{\pi}{5}$, and shade the area between them.

Would you like me to generate the sketch for you or continue with further details?

Related Questions:

1. How do you calculate the area enclosed between two curves?
2. What is the period of the function $y = 5 \cos(5x)$?
3. How do you handle a trigonometric equation like $\cos(5x) = \frac{1}{2}$?
4. What is the significance of points of intersection when calculating the enclosed area?
5. How would you generalize the process of finding the enclosed area for more complex functions?

Tip:

Always check the points of intersection before proceeding with further analysis such as area calculations.