To evaluate the integral

$\int_b^0 f(x) \, dx$

we'll interpret the shaded areas in the graph with the understanding of definite integrals and how they correspond to the area under a curve. Let’s go over the given information in the graph:

1. The areas bounded by the function $y = f(x)$ and the $x$-axis are labeled with their values:

   • The area between $0$ and $a$ is $16$.
   • The area between $a$ and $b$ is $3$ (in a triangular region below the $x$-axis).
   • The area between $b$ and $c$ is $13$.

2. When dealing with integrals of functions where regions are below the $x$-axis, those regions will contribute negative values to the integral.

Since we are asked to find $\int_b^0 f(x) \, dx$, let's break it down:

Step-by-Step Solution:

1. Integral from $b$ to $a$: $\int_b^a f(x) \, dx = -3$ This area is below the $x$-axis, so it’s negative.

2. Integral from $a$ to $0$: $\int_a^0 f(x) \, dx = 16$ This area is above the $x$-axis, so it’s positive.

3. Combine the Integrals: By the properties of integrals, we can add these segments: $\int_b^0 f(x) \, dx = \int_b^a f(x) \, dx + \int_a^0 f(x) \, dx = (-3) + 16 = 13$

Final Answer:

$\int_b^0 f(x) \, dx = 13$

Would you like further details on any step, or have additional questions?

Here are some related questions for further exploration:

1. What would be the result of $\int_0^b f(x) \, dx$?
2. How would we interpret $\int_a^c f(x) \, dx$ using the areas shown?
3. What changes if we calculate $\int_c^0 f(x) \, dx$?
4. How do we handle definite integrals when there are multiple regions below and above the $x$-axis?
5. What is the significance of switching limits in an integral, for example, $\int_0^b f(x) \, dx$ vs. $\int_b^0 f(x) \, dx$?

Tip: Remember that areas below the $x$-axis contribute negative values to the integral, while areas above the $x$-axis contribute positive values.