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.