To solve the given problem, we need to evaluate the definite integral $\int_{1}^{7} f(x) \, dx$ by calculating the net area between the graph of $f(x)$ and the $x$-axis over the interval $[1, 7]$.

Step 1: Analyze the graph

From the graph:

1. The interval $[1, 7]$ consists of two main segments:
   • From $x = 1$ to $x = 4$: This region is above the $x$-axis.
   • From $x = 4$ to $x = 7$: This region is below the $x$-axis.

Step 2: Calculate the area of each region

• Region 1 ($1 \leq x \leq 4$): This forms a triangle with base $4 - 1 = 3$ and height $6$. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 6 = 9$

• Region 2 ($4 \leq x \leq 7$): This forms a triangle below the $x$-axis with base $7 - 4 = 3$ and height $3$. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 3 = 4.5$

Since this area is below the $x$-axis, it contributes negatively to the integral.

Step 3: Compute the net area

The definite integral is given by: $\int_{1}^{7} f(x) \, dx = \text{Area of Region 1} - \text{Area of Region 2}$ $\int_{1}^{7} f(x) \, dx = 9 - 4.5 = 4.5$

Final Answer:

$\int_{1}^{7} f(x) \, dx = 4.5$

Let me know if you'd like more details or have further questions! Here are 5 related questions:

1. How do we calculate definite integrals for more complex piecewise functions?
2. What are the properties of definite integrals regarding symmetry?
3. How do you determine whether to subtract or add areas when computing integrals?
4. How can integrals be used to calculate the area under a curve with multiple intersections?
5. What is the significance of negative areas in integrals?

Tip: Always check whether the graph crosses the $x$-axis to accurately compute net areas.