To evaluate the integral $\int_{-3}^{3} f(x) \, dx$ from the graph, we need to calculate the area under the curve from $x = -3$ to $x = 3$.

Steps:

1. Divide the regions: The graph can be divided into three regions from $-3$ to $3$:

   • Region 1 (from $x = -3$ to $x = -1$): This area is above the x-axis.
   • Region 2 (from $x = -1$ to $x = 1$): This area is below the x-axis.
   • Region 3 (from $x = 1$ to $x = 3$): This area is above the x-axis.

2. Calculate the areas of each region:

   • Region 1: This appears to form a triangle with a base of $2$ (from $x = -3$ to $x = -1$) and a height of $3$. The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. $\text{Area of Region 1} = \frac{1}{2} \times 2 \times 3 = 3$
   • Region 2: This is a semicircle with a diameter of $2$ (from $x = -1$ to $x = 1$) and a radius of $1$. The area of a semicircle is $\frac{1}{2} \pi r^2$. $\text{Area of Region 2} = \frac{1}{2} \pi \times (1)^2 = \frac{\pi}{2}$ Since this area is below the x-axis, it is considered negative: $\text{Area of Region 2} = -\frac{\pi}{2}$
   • Region 3: This forms another triangle with a base of $2$ (from $x = 1$ to $x = 3$) and a height of $3$. $\text{Area of Region 3} = \frac{1}{2} \times 2 \times 3 = 3$

3. Sum the areas: $\int_{-3}^{3} f(x) \, dx = \text{Area of Region 1} + \text{Area of Region 2} + \text{Area of Region 3} = 3 - \frac{\pi}{2} + 3$ $\int_{-3}^{3} f(x) \, dx = 6 - \frac{\pi}{2}$

Final Answer:

$\int_{-3}^{3} f(x) \, dx = 6 - \frac{\pi}{2}$

Would you like further clarification on any of these steps?

Here are some related questions to expand on this topic:

1. How do we evaluate integrals using different methods, such as Riemann sums?
2. What is the significance of negative areas in definite integrals?
3. How do we approximate the area of irregular shapes under curves?
4. Can definite integrals have applications in physics, such as finding work done?
5. How do we interpret integrals when the curve crosses the x-axis multiple times?

Tip: When evaluating an integral graphically, always pay attention to whether the areas are above or below the x-axis, as this affects their sign in the integral calculation.