To solve the problem, we need to analyze the graph of the function $f(x)$ and evaluate the given definite integrals over the specified intervals. The question asks to arrange the integrals in increasing order.

Step-by-Step Analysis:

1. Graph Overview:

   • The graph is of a continuous function $f(x)$ that starts negative, increases to a positive peak, decreases, and then has a positive region again.
   • Important points to note are the zeros and the behavior of the function in different intervals.

2. Intervals and Integrals:

   • A: $\int_{0}^{1} f(x) \, dx$
   • B: $\int_{0}^{2} f(x) \, dx$
   • C: $\int_{0}^{3} f(x) \, dx$
   • D: $\int_{1}^{3} f(x) \, dx$

3. Understanding the Integrals:

   • $\int_{0}^{1} f(x) \, dx$ (A): This integral calculates the area under the curve from $x = 0$ to $x = 1$.
     • From the graph, it appears $f(x)$ is negative in this region, so this integral will yield a negative value.
   • $\int_{0}^{2} f(x) \, dx$ (B): This integrates from $x = 0$ to $x = 2$.
     • The graph shows $f(x)$ is negative from $x = 0$ to somewhere between 0 and 1, and positive from there to 2. The net area should be close to zero, potentially slightly positive or negative.
   • $\int_{0}^{3} f(x) \, dx$ (C): This integral takes into account the area from $x = 0$ to $x = 3$.
     • From $x = 2$ to $x = 3$, the function seems to be mostly positive, so this integral should give a positive value.
   • $\int_{1}^{3} f(x) \, dx$ (D): This is the area under the curve from $x = 1$ to $x = 3$.
     • Here, $f(x)$ is positive, so this integral will yield a positive value as well.

Determining the Order:

• Since $A$ is negative (smallest value), $B$ is around zero or slightly positive, $C$ is positive but smaller than $D$, and $D$ is the largest, the order is:

$\text{Order: A (smallest) < B < C < D (largest)}$

Final Answer:

The correct order from smallest to largest is: $A < B < C < D$

Would you like any more details or have any other questions?

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Here are some follow-up questions that might interest you:

1. How do we calculate definite integrals using the area under a curve?
2. What is the significance of the sign of an integral (positive or negative)?
3. How would the order change if the function was negative over a different interval?
4. Can the integral of a function over a range ever be zero? When does that happen?
5. How do we interpret the integral of a function that crosses the x-axis?
6. What are some numerical methods to estimate integrals?
7. How does the Fundamental Theorem of Calculus relate to definite integrals?
8. What is the relationship between indefinite integrals and definite integrals?

Tip: When analyzing the integral of a function over an interval, focus on the sign and magnitude of the function over that interval to estimate the integral value.