To estimate $\int_0^9 f(t) \, dt$ from the graph, we look at the approximate area under the curve $f(t)$ from $t = 0$ to $t = 9$. Here's how to proceed:

1. Shape Approximation: The curve appears to rise non-linearly from 0 to about 3 at $t = 9$. It roughly forms a triangular or trapezoidal region.
2. Area Estimate:
   • If approximating with a triangle, the base is 9, and the height is approximately 3: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 3 = 13.5.$
   • Slight overestimation for a triangle matches the closest value: 13.
3. The best estimate for $\int_0^9 f(t) \, dt$ is 13.

Let me know if you'd like further clarification! Here are some additional exploration questions:

1. How does the accuracy of this estimate depend on the method used (triangular vs. trapezoidal)?
2. What alternative numerical integration methods could be applied to refine the estimate?
3. Could a midpoint or Simpson's rule provide more accuracy for this problem?
4. What impact does the curve's non-linearity have on the precision of approximating its integral?
5. How can we generalize this approach to approximate any similar integral graphically?

Tip: When estimating areas under a curve, breaking the region into simpler shapes (like rectangles, triangles, or trapezoids) can provide a quick, intuitive approximation!