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). 7 (Simplify your answer.) c. \( \int_{2}^{7} 9 g(x) d x=72 \) (Simplify your answer.) d. \( \int_{4}^{7} f(x) d x=\square \) (Simplify your answer.)

To solve for the required integrals, let’s break down the provided information step by step. First, we have \( \int_{2}^{4} f(x) dx = -3 \) and \( \int_{2}^{7} f(x) dx = 7 \). **c.** To find \( \int_{2}^{7} 9 g(x) dx \), we can factor out the constant 9 from the integral: \[ \int_{2}^{7} 9 g(x) dx = 9 \int_{2}^{7} g(x) dx = 9 \cdot 8 = 72. \] So the answer for part c is **72**. **d.** To find \( \int_{4}^{7} f(x) dx \), we can use the property of definite integrals: \[ \int_{4}^{7} f(x) dx = \int_{2}^{7} f(x) dx - \int_{2}^{4} f(x) dx. \] Substituting the known values, we have: \[ \int_{4}^{7} f(x) dx = 7 - (-3) = 7 + 3 = 10. \] Thus, the answer for part d is **10**.