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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.

Compute the integral \( \int_{-3}^{13} f(x) \, dx \) where the graph of \( f(x) \) is the red curve given in the figure.

This problem involves computing the integral of a piecewise function, combining a linear segment and a semi-circular arc. Learn how to determine the area under each section, apply the formula for the area of a semi-circle, and use integral calculus to find the exact answer: 3.5 + 8π.

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