To solve this problem, we need to compute the definite integral of $f(t)$ from 4 to the specified values of $x$, based on the given piecewise graph.

The graph is divided into several segments where the value of $f(t)$ is either constant or linear. We'll calculate the integral $A(x) = \int_4^x f(t) \, dt$ by finding the area under the curve for each segment between $t = 4$ and the target $x$.

Step 1: Analyzing the graph

Segment 1: From $t = 4$ to $t = 5$

• The graph of $f(t)$ is a straight line increasing linearly from $f(4) = -3$ to $f(5) = 3$.
• This area is a trapezoid with:
  • Height (base length): $5 - 4 = 1$
  • Bases: $|f(4)| = 3$ and $|f(5)| = 3$

$\text{Area} = \frac{1}{2} \times (3 + 3) \times 1 = 3$

Segment 2: From $t = 5$ to $t = 9$

• $f(t)$ is constant at $f(t) = 3$.
• Area under this segment:
  • Height: $9 - 5 = 4$
  • Constant value: $3$

$\text{Area} = 3 \times 4 = 12$

Step 2: Compute $A(5)$ and $A(9)$

For $A(5)$:

The integral accumulates the area only from $t = 4$ to $t = 5$: $A(5) = \int_4^5 f(t) \, dt = 3$

For $A(9)$:

The integral accumulates the area from $t = 4$ to $t = 9$, summing both segments: $A(9) = \int_4^5 f(t) \, dt + \int_5^9 f(t) \, dt = 3 + 12 = 15$

Final Results:

$A(5) = 3, \quad A(9) = 15$

Would you like a breakdown of specific steps or have additional questions? Here are five related questions for further exploration:

Related Questions:

1. How does the definite integral change when the lower limit is shifted from 4 to another value?
2. What is the significance of positive and negative areas under the graph in integrals?
3. How would you calculate the integral if the graph contained a curve instead of linear segments?
4. What would $A(x)$ look like as a function (graph of $A(x)$) for all $x$ in this range?
5. How does the Fundamental Theorem of Calculus apply to piecewise linear graphs?

Tip:

When computing areas under piecewise linear graphs, break the graph into manageable segments and carefully account for positive and negative contributions to the integral!