To solve for $\int_a^c f(x) \, dx$, we can use the property of definite integrals that allows us to add integrals over adjacent intervals. Specifically:

$\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx$

From the given information:

• $\int_a^b f(x) \, dx = 9$
• $\int_b^c f(x) \, dx = -7$

Substituting these values into the equation:

$\int_a^c f(x) \, dx = 9 + (-7) = 2$

Thus, $\int_a^c f(x) \, dx = 2$.

Would you like further clarification on integral properties?

Here are 5 related questions to expand this topic:

1. How would the result change if $\int_b^c f(x) \, dx$ were a positive value?
2. Can you calculate $\int_a^b f(x) \, dx + \int_a^c f(x) \, dx$?
3. What if $\int_a^b f(x) \, dx = 0$ and $\int_b^c f(x) \, dx = 0$, what does that tell us about the function $f(x)$?
4. How would you interpret the result geometrically on the graph of $f(x)$?
5. What is the fundamental theorem of calculus, and how does it relate to this problem?

Tip: When integrating over adjacent intervals, always make sure the boundaries match up properly, so you can apply the additive property of integrals.