Question

use the midpoint rule to approximate the integral

Answer 1

\[\int\limits_{-7}^{-1}(-5x-9x ^{2})timesdx\]

Answer 2

with n=3

Answer 3

Because n = 3, we divide the interval into three equal subintervals: \[[-7,-1] = [-7,-5] \cup [-5,-3] \cup [-3,-1].\]
In each of these subintervals we approximate the integral to the length of the subinterval times the value of the function at the midpoint of the interval. Therefore,

\[I \doteq \int_{-7}^{-1}\underbrace{(-5x - 9x^2)}_{f(x)} dx \approx 2 \cdot f(-6) + 2 \cdot f(-4) + 2 \cdot f(-2). \]This finally yields \[I \approx -888\] (which is near the exact result, -906).