To solve the integral $\int_{3}^{13} f(x) \, dx$, we need to analyze the graph of $f(x)$ from $x = 3$ to $x = 13$, which is the area under the red curve within these bounds.

Step-by-Step Solution:

1. Identify regions in the graph:

   • From $x = 3$ to $x = 7$, the red curve is a line sloping upward.
   • From $x = 7$ to $x = 13$, the curve forms a semicircle.

2. Calculate the area from $x = 3$ to $x = 7$:

   • This segment appears to form a right triangle with a base of $4$ units (from $x = 3$ to $x = 7$) and a height of $6$ units.
   • The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$.
   • So, $\text{Area}_{\text{triangle}} = \frac{1}{2} \times 4 \times 6 = 12$.

3. Calculate the area from $x = 7$ to $x = 13$:

   • This segment is a semicircle with a diameter of $6$ (from $x = 7$ to $x = 13$), so its radius is $3$.
   • The area of a semicircle is $\frac{1}{2} \pi r^2$.
   • So, $\text{Area}_{\text{semicircle}} = \frac{1}{2} \pi \times 3^2 = \frac{9\pi}{2}$.

4. Combine the areas:

   • Total area $= 12 + \frac{9\pi}{2}$.

Final Answer:

$\int_{3}^{13} f(x) \, dx = 12 + \frac{9\pi}{2}$

Would you like more details, or do you have any questions?

Here are some related questions to consider:

1. What would happen if the semicircle was flipped to be below the x-axis?
2. How would the integral change if the bounds were from $x = 0$ to $x = 13$?
3. What is the interpretation of the integral result in terms of net area?
4. How would you calculate the integral if the curve had a different shape in each segment?
5. How does changing the radius of the semicircle affect the area calculation?

Tip: When working with integrals of areas under curves, sketching shapes like triangles or semicircles can simplify area calculations!