The task is to compute the integral $\int_{3}^{13} f(x) \, dx$, where the function $f(x)$ is represented by the red curve in the provided graph.

Approach:

1. Analyze the Graph: The curve seems to have distinct regions where it either forms a straight line or a part of a circle.
2. Break the Integral into Parts: Since we have different behaviors in the curve from $x = 3$ to $x = 13$, we can split the integral into parts corresponding to these behaviors.

Step-by-Step Breakdown:

• From $x = 3$ to $x = 8$, the curve forms a straight line segment.

  • This region forms a trapezoid, and the area under the curve can be computed geometrically.

• From $x = 8$ to $x = 13$, the curve forms a circular arc.

  • The area of this portion can be computed as part of a sector of a circle or derived using the equation of the circle (if known).

I will calculate each part separately and sum them up to obtain the final answer.

First Segment: Linear Part (from $x = 3$ to $x = 8$)

• The curve seems to form a trapezoid, so the area is: $A_1 = \frac{1}{2} \times ( \text{Base 1} + \text{Base 2}) \times \text{Height}$ where:

  • Base 1: height at $x = 3$, which is $2$,
  • Base 2: height at $x = 8$, which is $4$,
  • Height: $8 - 3 = 5$.

  Substituting these values: $A_1 = \frac{1}{2} \times (2 + 4) \times 5 = 15$

Second Segment: Circular Arc (from $x = 8$ to $x = 13$)

• The shape suggests part of a circle with a radius of $4$ (since the curve rises from $y = 4$ to $y = 8$).
• To compute the area under this circular arc, we can use the formula for the area of a sector: $A_2 = \frac{\theta}{2\pi} \times \pi r^2$ where $\theta$ is the central angle in radians, and $r = 4$ is the radius of the circle. The length of the arc corresponds to half of the circle (180° or $\pi$ radians), so: $A_2 = \frac{\pi}{2\pi} \times \pi (4^2) = 8\pi$

Thus, the total area under the curve, which is the value of the integral, is: $A = A_1 + A_2 = 15 + 8\pi$

Final Answer:

$\boxed{15 + 8\pi}$

Would you like further clarification on any step? Here are 5 related questions to explore:

1. How would the solution change if the curve had a different radius for the circular segment?
2. Can you describe the relationship between definite integrals and the area under curves?
3. What is the geometric interpretation of integration in calculus?
4. How do we approximate areas under irregular curves that don’t have simple geometric shapes?
5. How can the trapezoidal rule be used to approximate this integral numerically?

Tip: Always break down complex integrals into smaller, more manageable segments, especially when the function's behavior changes in different intervals.