Math 110 Course Resources - Definite Integrals Course Packet on substitution for definite integrals \( \int_{21}^{56} f(x) d x=13 \), then compute the value of the following definite integral. \[ \int_{3}^{8} f(7 x)+4 d x \] x

What is the average rate of change on the function \( f(x)=x^{2} \) between \( x= \) 1 and \( x=3 \) ? -1

\( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)

Suppose \( \int^{4} f(x) d x=6, \int^{7} f(x) d x=-2 \), and \( \int^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{2}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) 7 \( \int_{2}[g(x)-f(x)] d x=6 \) 2 (Simplify your answer.) 7 \( \int_{2}[5 g(x)-f(x)] d x=\square \) (Simplify your answer.)

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{2}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

The president of a company that manufactures toy cars thinks that the function \( P(c)=-2 c^{2}+14 c-20 \) represents the company's profit, where \( c \) is the number of cars produced, in thousands, and \( P(c) \) is the company's profit, in hundreds of thousands of dollars. Determine the maximum profit the company can earn.