The problem asks to evaluate the integral of $g(x)$ from 2 to 4 ($\int_{2}^{4} g(x) \, dx$), using the given values:

• $\int_{2}^{9} f(x) \, dx = 11$
• $\int_{2}^{9} g(x) \, dx = 7$
• $\int_{4}^{9} f(x) \, dx = 7$
• $\int_{2}^{4} g(x) \, dx = 3$

Solution:

We are directly given: $\int_{2}^{4} g(x) \, dx = 3$

Thus, the answer is:

$\boxed{3}$

Would you like me to help solve any follow-up parts or explore other related integrals?

Here are 5 questions to deepen understanding:

1. How would you calculate $\int_{4}^{9} g(x) \, dx$ if needed?
2. Can you verify $\int_{2}^{9} g(x) \, dx$ using smaller intervals?
3. How does splitting an integral into subintervals work mathematically?
4. Why do the bounds of integration matter in definite integrals?
5. What is the connection between definite integrals and the Fundamental Theorem of Calculus?

Tip: Always check if the integral can be split into smaller intervals based on given data.