To solve the problem, we need to calculate the definite integral of $f(x)$ from $0$ to $2$ based on the graph of $f(x)$. The integral can be interpreted as the net area under the curve between the given bounds.

Looking at the graph in your uploaded image:

1. The function $f(x)$ includes both positive and negative areas within the bounds $[0, 2]$.
2. To find the integral, calculate the area of each segment individually:
   • A triangle in the positive region (above the $x$-axis) contributes positively to the integral.
   • A triangle in the negative region (below the $x$-axis) contributes negatively to the integral.

Step-by-Step Calculation

1. Positive Triangle (from $0$ to $1$):

• Base: $1$ unit (from $x = 0$ to $x = 1$).
• Height: $2$.
• Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 2 = 1$.

2. Negative Triangle (from $1$ to $2$):

• Base: $1$ unit (from $x = 1$ to $x = 2$).
• Height: $-2$ (negative because it's below the $x$-axis).
• Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times (-2) = -1$.

Total Integral

• The total integral is the sum of the areas: $\int_0^2 f(x) \, dx = 1 + (-1) = 0.$

Final Answer

$\int_0^2 f(x) \, dx = 0$

Let me know if you'd like further clarification or more details on this process!

Here are five related questions for practice:

1. How do you handle cases where the curve is not linear but involves curved segments?
2. How would you find the integral if the bounds were changed to $[0, 3]$?
3. What is the significance of the integral being zero in this case?
4. How does the concept of net area apply to integrals in practical scenarios?
5. How can definite integrals be used to compute physical quantities, such as distance or work?

Tip: Always pay close attention to whether the function is above or below the $x$-axis to determine the sign of the integral!