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The problem involves evaluating two integrals based on the graph, where the given function \( y = f(x) \) is composed of two semicircles. The integrals to evaluate are:

\[
\int_0^3 f(x) \, dx \quad \text{and} \quad \int_3^6 f(x) \, dx
\]

### Step 1: Analyzing the Graph
1. The graph consists of two semicircles:
   - The first semicircle, from \( x = 0 \) to \( x = 3 \), is below the x-axis.
   - The second semicircle, from \( x = 3 \) to \( x = 6 \), is above the x-axis.
   
2. Since both parts are semicircles, we can calculate the area for each part using the formula for the area of a semicircle:
   \[
   \text{Area of a semicircle} = \frac{1}{2} \pi r^2
   \]
   where \( r \) is the radius of the semicircle.

### Step 2: Evaluating the Integrals
1. **First integral** \( \int_0^3 f(x) \, dx \):
   - This corresponds to the area of the first semicircle (below the x-axis) with a radius of 1.5. Since it's below the axis, the integral will give a negative value.
   - Radius \( r = \frac{3}{2} = 1.5 \).
   \[
   \int_0^3 f(x) \, dx = - \frac{1}{2} \pi (1.5)^2 = - \frac{9\pi}{8}
   \]

2. **Second integral** \( \int_3^6 f(x) \, dx \):
   - This corresponds to the area of the second semicircle (above the x-axis) with a radius of 1.5.
   \[
   \int_3^6 f(x) \, dx = \frac{1}{2} \pi (1.5)^2 = \frac{9\pi}{8}
   \]

### Step 3: Final Answer
The total integral from \( 0 \) to \( 6 \) is the sum of the two parts:
\[
\int_0^6 f(x) \, dx = \left( - \frac{9\pi}{8} \right) + \left( \frac{9\pi}{8} \right) = 0
\]

Thus, the value of the integral is \( 0 \).

Would you like more details on this solution, or do you have any other questions?

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Here are 5 related questions to deepen your understanding:
1. How do you calculate the area of a full circle using integration?
2. What is the difference between the geometric and analytic approach to evaluating an integral?
3. How would the answer change if one of the semicircles were not perfectly shaped?
4. What happens if a function is neither positive nor negative in an integral region?
5. How does the concept of symmetry simplify evaluating definite integrals?

**Tip:** When dealing with areas under curves, always pay attention to whether the area lies above or below the x-axis to determine the sign of the integral.

Evaluate the integrals \( \int_0^3 f(x) \, dx \) and \( \int_3^6 f(x) \, dx \) based on the graph where the two parts of the graph are semicircles.

This problem evaluates two integrals based on the geometry of semicircles from 0 to 6. The solution involves finding the area under the curve using the formula for the area of a semicircle and applying the Fundamental Theorem of Calculus.

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