Use the midpoint rule to approximate the integral 6x-9x^2 from 2 to 11 with n=3

To approximate the integral using the midpoint rule, follow these steps:

1. Determine the width of each subinterval: Since n = 3, divide the total interval (11 - 2) into 3 equal subintervals. The width of each subinterval (Δx) is calculated by (b - a) / n, where "b" is the upper limit of integration (11) and "a" is the lower limit of integration (2). In this case, Δx = (11 - 2) / 3 = 3.

2. Find the midpoint of each subinterval: To find the midpoint of each subinterval, use the formula (a + (i - 1/2) * Δx), where "i" represents the subinterval number (from 1 to n). In this case, i = 1, 2, and 3. By plugging in the values, we get: x_1 = 2 + (1 - 1/2) * 3 = 3.5, x_2 = 2 + (2 - 1/2) * 3 = 6.5, and x_3 = 2 + (3 - 1/2) * 3 = 9.5.

3. Evaluate the function at each midpoint: Plug each midpoint value (x_1, x_2, and x_3) into the integrand, which is 6x - 9x^2. Calculate the value for each midpoint:
- f(x_1) = 6 * 3.5 - 9 * (3.5)^2 = -38.25
- f(x_2) = 6 * 6.5 - 9 * (6.5)^2 = -180.25
- f(x_3) = 6 * 9.5 - 9 * (9.5)^2 = -358.75

4. Calculate the sum of the function values: Add up the values of the function at each midpoint. In this case, -38.25 + (-180.25) + (-358.75) = -577.25.

5. Multiply the sum by the width of each subinterval: Multiply the sum from step 4 by the width of each subinterval (Δx = 3), then divide by n = 3. This gives us:
-577.25 * 3 / 3 = -577.25

Therefore, the approximated value of the integral of 6x - 9x^2 from 2 to 11 using the midpoint rule with n = 3 is -577.25.