Use the Midpoint Rule to approximate the integral:
the upper number of the integral is 14 the lower number is 10 (4x+6x2)dx. with n=3
To approximate the integral using the Midpoint Rule with n = 3, we divide the interval [10, 14] into n subintervals of equal length. Since n = 3, we will have 3 subintervals.
The width of each subinterval is given by Δx = (b - a) / n, where b is the upper limit (14) and a is the lower limit (10).
Δx = (14 - 10) / 3 = 4 / 3 = 1.33333
Next, we need to find the midpoint of each subinterval. The midpoint of each subinterval can be calculated using the formula:
xi = a + (i - 0.5)Δx
where i represents the subinterval number (1, 2, 3).
For i = 1:
x1 = 10 + (1 - 0.5) * 1.33333
= 10 + 0.5 * 1.33333
= 10 + 0.666665
= 10.666665
For i = 2:
x2 = 10 + (2 - 0.5) * 1.33333
= 10 + 1.5 * 1.33333
= 10 + 2.000005
= 12
For i = 3:
x3 = 10 + (3 - 0.5) * 1.33333
= 10 + 2.5 * 1.33333
= 10 + 3.333325
= 13.333325
Now, we evaluate the function (4x + 6x^2) at each midpoint and approximate the integral by summing up the areas of the rectangles. The area of each rectangle is given by the function value multiplied by the width (Δx).
Let's calculate the function values at each midpoint:
f(x1) = 4 * 10.666665 + 6 * (10.666665)^2
≈ 45.33334 + 726.6667
≈ 771
f(x2) = 4 * 12 + 6 * (12)^2
= 48 + 864
= 912
f(x3) = 4 * 13.333325 + 6 * (13.333325)^2
≈ 53.3333 + 710.2222
≈ 764.5555
Finally, we can approximate the integral by summing up the areas of the rectangles:
Approximation ≈ f(x1) * Δx + f(x2) * Δx + f(x3) * Δx
≈ 771 * 1.33333 + 912 * 1.33333 + 764.5555 * 1.33333
≈ 1028.6667 + 1215.9996 + 1019.4074
≈ 3264.0737
Therefore, the Midpoint Rule approximation of the integral of (4x + 6x^2) from 10 to 14 with n = 3 is approximately equal to 3264.0737.