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To solve this problem, we need to compute the integral

\[
\int_3^{13} f(x) \, dx
\]

where the graph of \( f(x) \) is given. The area under this curve, bounded between \( x = 3 \) and \( x = 13 \), represents the integral. From the graph, we can see that the curve is composed of two distinct sections:

1. **A straight line** from \( x = 3 \) to \( x = 6 \).
2. **A semicircle** with a radius of 4 centered at \( x = 9 \) and spanning from \( x = 5 \) to \( x = 13 \).

We will calculate the area under each section separately.

### Step 1: Area under the Line Segment from \( x = 3 \) to \( x = 5 \)
The line segment from \( x = 3 \) to \( x = 5 \) has a triangular shape. It starts at the point \( (3, 0) \) and rises linearly to \( (5, 4) \). The area of this triangle can be calculated as:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (5 - 3) \times 4 = \frac{1}{2} \times 2 \times 4 = 4
\]

### Step 2: Area under the Semicircle from \( x = 5 \) to \( x = 13 \)
The semicircle extends horizontally from \( x = 5 \) to \( x = 13 \) with a radius of 4, centered at \( x = 9 \). The area of a full circle with radius \( r = 4 \) would be:

\[
\text{Area of full circle} = \pi r^2 = \pi \times 4^2 = 16\pi
\]

Since we only need the area of the upper half (a semicircle), we take half of this value:

\[
\text{Area of semicircle} = \frac{1}{2} \times 16\pi = 8\pi
\]

### Step 3: Total Area under \( f(x) \) from \( x = 3 \) to \( x = 13 \)
Adding the areas from the line segment and the semicircle gives:

\[
\int_3^{13} f(x) \, dx = 4 + 8\pi
\]

### Final Answer
The exact answer to the integral is:

\[
\int_3^{13} f(x) \, dx = 4 + 8\pi
\]

Would you like further details or have any questions?

---

### Follow-Up Questions

1. How would the answer change if the semicircle were below the \( x \)-axis?
2. What would happen if we integrated from \( x = 0 \) to \( x = 13 \)?
3. How do we interpret the integral geometrically if the function changes below and above the \( x \)-axis?
4. How would we approach this if the curve included multiple semicircles?
5. What is the importance of symmetry in calculating areas for shapes like semicircles?

### Tip
When interpreting areas under curves for integrals, splitting the region based on shapes (like triangles and semicircles) often simplifies the calculation.

Compute the integral \( \int_3^{13} f(x) \, dx \), where the graph of \( f \) is the red curve given in the figure. Note: On the graph of \( f \), all points that are relevant for this computation have integer coordinates; the blue line is just for reference and is not the \( x \)-axis.

This problem involves calculating the integral of a piecewise function composed of a line segment and a semicircle. The solution includes finding areas of both shapes separately and using integral properties. Ideal for students learning Calculus I.

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