Use the Midpoint Rule with n =3 to approximate the integral
�ç−1−8(10x−9x2)dx
To approximate the integral �ç−1−8(10x−9x^2)dx using the Midpoint Rule with n = 3, we need to follow these steps:
Step 1: Determine the width of each subinterval.
The width of each subinterval is given by Δx, which is calculated by dividing the length of the interval by the number of subintervals:
Δx = (b - a) / n
where a and b are the lower and upper limits of the interval respectively, and n is the number of subintervals. In this case, a = -1, b = -8, and n = 3. So we have:
Δx = (-8 - (-1)) / 3 = -7 / 3.
Step 2: Calculate the midpoint of each subinterval.
To calculate the midpoint of each subinterval, we use the formula:
x_i = a + (i - 0.5)Δx
where i ranges from 1 to n. In this case, n = 3. So we have:
x_1 = -1 + (1 - 0.5)(-7 / 3)
x_2 = -1 + (2 - 0.5)(-7 / 3)
x_3 = -1 + (3 - 0.5)(-7 / 3)
Step 3: Evaluate the function at each midpoint.
Now, substitute each midpoint value (x_i) into the function f(x) = -8(10x - 9x^2) and calculate the function value at each midpoint:
f(x_1) = -8(10(-1 + (1 - 0.5)(-7 / 3)) - 9(-1 + (1 - 0.5)(-7 / 3))^2)
f(x_2) = -8(10(-1 + (2 - 0.5)(-7 / 3)) - 9(-1 + (2 - 0.5)(-7 / 3))^2)
f(x_3) = -8(10(-1 + (3 - 0.5)(-7 / 3)) - 9(-1 + (3 - 0.5)(-7 / 3))^2)
Step 4: Calculate the approximate integral.
Finally, we can calculate the approximate integral by summing up the product of the function values at each midpoint and the width of each subinterval:
Approximate integral ≈ Δx * (f(x_1) + f(x_2) + f(x_3))
= -7/3 * (f(x_1) + f(x_2) + f(x_3))
You can now substitute the values of x_i and f(x_i) into the above equation to find the approximate integral.