The problem asks to compute the integral $\int_{-3}^{13} f(x) \, dx$ where the graph of $f(x)$ is represented by the red curve, and all relevant points have integer coordinates.

From the graph, we can identify that the function $f(x)$ is a piecewise function composed of:

1. A linear segment from $x = -3$ to $x = 4$.
2. A semi-circular arc from $x = 4$ to $x = 12$.
3. A constant value from $x = 12$ to $x = 13$.

Step-by-step breakdown:

1. Linear segment from $x = -3$ to $x = 4$: The line passes through points $(-3, -3)$ and $(4, 4)$. The equation of this line can be determined using the slope-intercept form: $f(x) = x$ Hence, the integral over this interval is: $\int_{-3}^{4} x \, dx = \left[ \frac{x^2}{2} \right]_{-3}^{4} = \frac{4^2}{2} - \frac{(-3)^2}{2} = 8 - 4.5 = 3.5$

2. Semi-circular arc from $x = 4$ to $x = 12$: This section represents a semi-circle with radius 4 (centered at $(8, 4)$) and height 4. The area of a semi-circle is: $\text{Area} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4)^2 = 8\pi$ Since we are calculating an integral, we directly add the area under the curve: $\int_{4}^{12} f(x) \, dx = 8\pi$

3. Constant segment from $x = 12$ to $x = 13$: The function has a constant value of $f(x) = 0$, so: $\int_{12}^{13} 0 \, dx = 0$

Total Integral:

Combining all contributions, the total integral is: $\int_{-3}^{13} f(x) \, dx = 3.5 + 8\pi + 0 = 3.5 + 8\pi$

So, the exact answer is: $\boxed{3.5 + 8\pi}$

Let me know if you would like more details or have further questions!

Here are 5 questions related to this problem:

1. How do you derive the equation of a line between two points?
2. What is the formula for the area of a circle and how is it applied in integrals?
3. How would the integral change if the semi-circle were a full circle?
4. What are the rules for integrating piecewise functions?
5. Can you find the integral using numerical methods if the shape was more complex?

Tip: For semi-circles and other geometric shapes, converting the area into an integral helps break complex problems into simpler parts.