To evaluate the definite integrals $\int_{0}^{4} g(t) \, dt$ and $\int_{3}^{5} g(t) \, dt$ using the graph of $y = g(t)$, let's examine the regions under the graph.

Step 1: Understanding the Graph

The graph is composed of line segments forming a triangular shape, making it piecewise linear. We need to find the area under $g(t)$ from 0 to 4 and then from 3 to 5, noting whether the area is above or below the x-axis.

Step 2: Evaluating $\int_{0}^{4} g(t) \, dt$

1. From $t = 0$ to $t = 2$, the graph is above the x-axis, forming a triangle with base 2 and height 2.

   • Area of triangle from 0 to 2: $\frac{1}{2} \times 2 \times 2 = 2$.

2. From $t = 2$ to $t = 4$, the graph is also a triangle but below the x-axis with base 2 and height -2.

   • Area of triangle from 2 to 4: $\frac{1}{2} \times 2 \times (-2) = -2$.

The total area for $\int_{0}^{4} g(t) \, dt$ is: $2 + (-2) = 0.$

Step 3: Evaluating $\int_{3}^{5} g(t) \, dt$

1. From $t = 3$ to $t = 5$, the graph forms a triangle below the x-axis with base 2 and height -2.
   • Area of triangle from 3 to 5: $\frac{1}{2} \times 2 \times (-2) = -2$.

Therefore, $\int_{3}^{5} g(t) \, dt = -2$.

Final Answers

• $\int_{0}^{4} g(t) \, dt = 0$
• $\int_{3}^{5} g(t) \, dt = -2$

Would you like further details or have any questions?

Here are five related questions to explore further:

1. How would the integral change if the region from $t = 2$ to $t = 4$ had a different height?
2. What would be the result if we evaluated $\int_{1}^{5} g(t) \, dt$?
3. How can we interpret these integrals in terms of net area versus total area?
4. How would the answer change if we used absolute values for the areas?
5. Can you derive the integral if the function $g(t)$ is represented as a piecewise function?

Tip: Always look at whether the area is above or below the x-axis in definite integrals, as it affects the sign of each segment.