To evaluate $g(2)$, where $g(x) = \int_{-2}^x f(t) \, dt$, we need to find the area under the curve $f(t)$ from $t = -2$ to $t = 2$.

Here is the step-by-step process:

Step 1: Analyze the graph

The graph of $f(t)$ is made of line segments. We will calculate the areas of the regions between $t = -2$ and $t = 2$, remembering:

• Areas above the $t$-axis contribute positively.
• Areas below the $t$-axis contribute negatively.

Step 2: Break the integral into sub-intervals

The interval from $t = -2$ to $t = 2$ is divided into the following regions based on the graph:

1. From $t = -2$ to $t = 0$: a trapezoid.
2. From $t = 0$ to $t = 2$: a rectangle and a triangle.

Step 3: Calculate each area

1. From $t = -2$ to $t = 0$:
   The trapezoid has bases of $y = 2$ and $y = 4$, with a width of 2.
   Area = $\frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{width} = \frac{1}{2} \times (2 + 4) \times 2 = 6$.

2. From $t = 0$ to $t = 2$:

   • The rectangle has a height of 4 and a width of 2.
     Area = $4 \times 2 = 8$.

Step 4: Add areas

The total area from $t = -2$ to $t = 2$ is: $g(2) = 6 + 8 = 14.$

Final Answer:

$g(2) = 14.$

Would you like a more detailed breakdown or further clarifications?

Related Questions:

1. What is the value of $g(0)$ using the same graph?
2. How does $g'(x)$ relate to $f(x)$ in this scenario?
3. How would you interpret $g(x)$ if $f(x)$ were negative?
4. What is the significance of the Fundamental Theorem of Calculus in this context?
5. How would you calculate $g(4)$ for this function?

Tip:

Always pay attention to the signs of the areas (positive above the axis, negative below) when dealing with definite integrals!