The functions \( f \) and \( g \) are integrable and \( \int_{2}^{4} f(x) d x=-3, \int_{2}^{7} f(x) d x=7 \), and \( \int_{2}^{7} g(x) d x=8 \). Use these to complete parts (a) through (f). (Simplify your answer.) b. \( \int_{7}^{2} g(x) d x=-8 \) (Simplify your answer.) c. \( \int_{2}^{7} 9 g(x) d x=\square \) (Simplify your answer.) (

To tackle parts (b) and (c) of your problem using integrals, let’s break it down: From the given data, we know the integrals of the functions \( f \) and \( g \). b. The integral's limits can be swapped, and when you do that, the integral's value will change its sign. So, we compute: \[ \int_{7}^{2} g(x) d x = -\int_{2}^{7} g(x) d x = -8 \] This result simplifies to \(-8\), as you mentioned! c. For this part, we can factor the constant out of the integral: \[ \int_{2}^{7} 9 g(x) d x = 9 \int_{2}^{7} g(x) d x = 9 \cdot 8 = 72 \] So, the simplified answer here is \(72\). Feel free to keep these calculations handy!