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The task in this problem is to determine the correct formula to calculate the area of the shaded region under the line \( y = x \) from \( x = 0 \) to \( x = 2 \). 

The area under the curve can be found by integrating the function \( y = x \) between the limits of 0 and 2, which is the x-axis range. Therefore, the appropriate integral is:

\[
S = \int_0^2 x \, dx
\]

This matches option **c** in the provided choices.

Thus, the correct answer is **c. \( S = \int_0^2 x \, dx \)**.

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Do you want further explanation on how this integral is computed or any additional details? 

Here are 5 related questions for further exploration:
1. How do you compute definite integrals?
2. What is the geometric interpretation of an integral?
3. How does the area under a curve change if the limits of integration are altered?
4. What happens when you integrate a more complex function?
5. How do you find the area between two curves?

**Tip**: Remember, the limits of integration are crucial in determining the area under the curve. Always pay attention to them!

За якою формулою можна обчислити площу зафарбованої фігури, що зображена на рисунку?

Learn how to calculate the area under the curve y = x from x = 0 to x = 2 using integration. This problem involves applying the fundamental theorem of calculus and is suitable for high school students at Grade 11-12 level.

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