Let's solve the integrals for parts (a) and (b) using the information provided.

Part (a)

We are given: $\int_{2}^{4} f(x) \, dx = 44$ We need to evaluate: $\int_{2}^{4} 5f(x) \, dx$ Since the integral of a constant times a function can be expressed as the constant times the integral of the function: $\int_{2}^{4} 5f(x) \, dx = 5 \times \int_{2}^{4} f(x) \, dx = 5 \times 44 = 220$

Part (b)

We are given: $\int_{4}^{7} g(x) \, dx = 22$ We need to evaluate: $\int_{4}^{7} -6g(x) \, dx$ Similarly, using the property of integrals, we can factor out the constant: $\int_{4}^{7} -6g(x) \, dx = -6 \times \int_{4}^{7} g(x) \, dx = -6 \times 22 = -132$

Final Answers

• (a) $\int_{2}^{4} 5f(x) \, dx = 220$
• (b) $\int_{4}^{7} -6g(x) \, dx = -132$

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How would you evaluate $\int_{2}^{7} (f(x) + g(x)) \, dx$ using the provided integrals?
2. What is the integral of a sum of functions over the same interval?
3. Can you explain why we can factor out constants in definite integrals?
4. What happens if the integrand is multiplied by a variable instead of a constant?
5. How do the properties of definite integrals relate to the linearity of integration?

Tip: Remember, the linearity of integration allows you to split sums and factor out constants, making it easier to handle complex integrals by breaking them down into simpler parts.