https://file.aiquickdraw.com/mathbot/file/1726613654190eml8dq0z.jpg

The image shows two functions: 
1. \( y = 8 \cos(x) \)
2. \( y = 8 - 8 \cos(x) \)

You are asked to find the area between these two curves over the interval \( 0 \leq x \leq \pi \).

The area between two curves, \( f(x) \) and \( g(x) \), over an interval \([a, b]\) is given by the formula:

\[
\text{Area} = \int_a^b \left[ f(x) - g(x) \right] \, dx
\]

Here:
- \( f(x) = 8 - 8 \cos(x) \)
- \( g(x) = 8 \cos(x) \)
- The limits of integration are \( 0 \) and \( \pi \).

Now, I will calculate the area between these two curves.The area between the two curves \( y = 8 \cos(x) \) and \( y = 8 - 8 \cos(x) \) over the interval \( 0 \leq x \leq \pi \) is \( 8\pi \) square units.

Do you need any further details or clarifications?

Here are five related questions to explore further:
1. How would the area change if the limits of integration were different?
2. What happens if we replace the cosine function with a sine function?
3. Can this technique be used to find the area between curves that aren't trigonometric?
4. How does the graph of the functions \( y = 8 \cos(x) \) and \( y = 8 - 8 \cos(x) \) look?
5. What is the significance of calculating the area between curves in practical applications?

**Tip**: When calculating areas between curves, always ensure that the function representing the upper curve is subtracted from the lower curve to avoid negative areas.

Find the area between the curves y = 8 cos(x) and y = 8 - 8 cos(x), 0 ≤ x ≤ π.

Learn how to calculate the area between the curves y = 8 cos(x) and y = 8 - 8 cos(x) over the interval 0 ≤ x ≤ π. This problem involves key calculus concepts like the integral of trigonometric functions and is suitable for advanced high school or college-level students.

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