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To set up the definite integral that represents the area of the shaded region under the curve \( f(x) = \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \), we use the following integral:

\[
\int_{0}^{\frac{\pi}{2}} \cos(x) \, dx
\]

This integral will yield the area under the curve \( f(x) = \cos(x) \) over the interval \( [0, \frac{\pi}{2}] \). Note that you are not required to evaluate it, just to set it up as above.

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Here are some related questions to deepen understanding:
1. How do you determine the limits of integration for a given region?
2. What is the geometric interpretation of integrating a trigonometric function like \( \cos(x) \)?
3. Why is the integral of \( \cos(x) \) over \( [0, \frac{\pi}{2}] \) equal to the area under the curve?
4. What would change if the function were \( \sin(x) \) instead of \( \cos(x) \)?
5. How does the fundamental theorem of calculus relate to finding the area under a curve?

**Tip:** When setting up integrals, always pay attention to the limits of integration as they define the specific region for the area calculation.

Set up a definite integral that yields the area of the region. \( f(x) = \cos(x) \), from \( x = 0 \) to \( x = \frac{\pi}{2} \).

Learn how to set up a definite integral to find the area under the curve \( f(x) = \cos(x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \). This problem covers calculus concepts, including definite integrals and trigonometric functions, suitable for students in Grades 11-12.

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