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We are given that \( f(x) \) is an even function defined in the interval \([-2, 2]\), meaning \( f(x) = f(-x) \). Based on this, we need to analyze the integrals and select the correct relation from the options.

### Key properties of even functions:
For an even function \( f(x) \), the following integral property holds:
\[
\int_{-a}^{a} f(x) \, dx = 2 \int_0^{a} f(x) \, dx
\]
This is because the area on the interval \([0, a]\) is the same as on \([-a, 0]\) due to the symmetry of even functions.

### Analysis of the options:

#### Option a:
\[
\int_0^2 f(x) \, dx = \frac{1}{2} \int_{-2}^2 f(x) \, dx
\]
Using the property of even functions, we know:
\[
\int_{-2}^2 f(x) \, dx = 2 \int_0^2 f(x) \, dx
\]
If we divide both sides by 2, we get:
\[
\int_0^2 f(x) \, dx = \frac{1}{2} \int_{-2}^2 f(x) \, dx
\]
This matches exactly with option a, making this the correct choice.

#### Option b:
It states that there is no relationship between the integrals, which is incorrect, as we have already found a clear relationship in option a.

#### Option c:
\[
\int_0^2 f(x) \, dx = -\int_{-2}^0 f(x) \, dx
\]
This does not hold for an even function. In fact, for an even function:
\[
\int_{-2}^0 f(x) \, dx = \int_0^2 f(x) \, dx
\]
Therefore, option c is incorrect.

### Conclusion:
The correct answer is **a**.

---

Do you want more details on how the integrals of even functions are derived?

Here are 5 related questions to further your understanding:

1. How would the result change if \( f(x) \) were an odd function?
2. What is the relationship between integrals of odd functions over symmetric intervals?
3. Can we generalize this result for any even function on a symmetric interval \([-a, a]\)?
4. How can we interpret the integral of an even function geometrically?
5. How do definite integrals behave when the limits of integration are swapped?

**Tip**: Always check the symmetry properties of functions before integrating over symmetric intervals.

Given that f(x) is an even function defined on [-2,2] such that f(x) = f(-x), select the correct relationship between the integrals from the options provided.

Explore how to solve integrals involving even functions over symmetric intervals. This problem illustrates key properties of even functions and shows how symmetry helps simplify definite integrals.

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