Use the midpoint rule to approximate the integral 6x-9x^2 from 2 to 11 with n=3
To use the midpoint rule to approximate the integral, follow these steps:
Step 1: Determine the width of each interval.
In this case, the interval is [2, 11], and n = 3, representing the number of subintervals. To find the width of each interval, divide the length of the interval by the number of subintervals:
Interval width = (11 - 2) / 3 = 3
Step 2: Determine the midpoints of the subintervals.
To find the midpoints, start with the left endpoint of the interval and keep adding the width until you reach the right endpoint. For n = 3, the midpoints are:
Midpoint 1 = 2 + 0.5 * 3 = 3.5
Midpoint 2 = 2 + 1.5 * 3 = 6.5
Midpoint 3 = 2 + 2.5 * 3 = 9.5
Step 3: Evaluate the function at each midpoint.
Evaluate the function 6x - 9x^2 at each midpoint calculated in Step 2:
f(3.5) = 6(3.5) - 9(3.5)^2 = 21 - 110.25 = -89.25
f(6.5) = 6(6.5) - 9(6.5)^2 = 39 - 354.75 = -315.75
f(9.5) = 6(9.5) - 9(9.5)^2 = 57 - 855.75 = -798.75
Step 4: Calculate the sum of the function values.
Add up the function values obtained in Step 3:
Sum = -89.25 + (-315.75) + (-798.75) = -1203.75
Step 5: Multiply the sum by the width of each interval.
Multiply the sum calculated in Step 4 by the interval width:
Approximation = -1203.75 * 3 = -3611.25
Therefore, the approximate value of the integral 6x - 9x^2 from 2 to 11 with n = 3 using the midpoint rule is -3611.25.