The problem asks to find the area between the curves $y = 13 \sin(x)$ and $y = 13 \cos(x)$ over the interval $[0, \pi/4]$.

To solve this:

1. The area between two curves $y_1(x)$ and $y_2(x)$ over an interval $[a, b]$ is given by: $A = \int_a^b |y_1(x) - y_2(x)| \, dx$ In this case, we take $y_1(x) = 13 \cos(x)$ and $y_2(x) = 13 \sin(x)$.

2. The integral setup becomes: $A = \int_0^{\pi/4} |13 \cos(x) - 13 \sin(x)| \, dx$ Since $\cos(x) \geq \sin(x)$ on the interval $[0, \pi/4]$, the absolute value can be removed, giving: $A = \int_0^{\pi/4} 13 (\cos(x) - \sin(x)) \, dx$

3. Now, let's compute the integral: $A = 13 \int_0^{\pi/4} (\cos(x) - \sin(x)) \, dx$

Let me calculate this for you.The area between the curves $y = 13 \sin(x)$ and $y = 13 \cos(x)$ over the interval $[0, \pi/4]$ is approximately $5.385$ (rounded to three decimal places).

Do you want further details or have any questions?

Here are 5 related questions to consider:

1. How would the area change if the interval was extended beyond $[0, \pi/4]$?
2. Can you apply the same method to find the area between other trigonometric functions?
3. What happens if the curves $y = \sin(x)$ and $y = \cos(x)$ intersect over the interval?
4. How would you compute the area if the functions were shifted vertically?
5. What is the geometric significance of the area between two sine or cosine waves?

Tip: When finding areas between curves, always pay attention to whether one function is greater than the other over the interval; this affects whether you need absolute values.