Use the Midpoint Rule to approximate the integral
The integral from 2 to 7 of (–1x–6x^2)dx
To approximate the integral using the Midpoint Rule, we first need to divide the interval [2, 7] into subintervals of equal width. Let's say we want to use n subintervals. The width of each subinterval is given by:
Δx = (b - a) / n
= (7 - 2) / n
= 5 / n
Next, we need to determine the midpoint of each subinterval. The midpoint of a subinterval is the average of its left and right endpoints. Since our subintervals have equal width, the left endpoint of each subinterval can be calculated using:
x_i = a + (i - 1)Δx
= 2 + (i - 1)(5 / n)
where i is the index of the subinterval, ranging from 1 to n.
Now, we can calculate the approximation of the integral using the Midpoint Rule formula:
∫(–1x–6x^2)dx ≈ Δx * Σf(x_i)
where f(x_i) is the function evaluated at the midpoint of each subinterval.
Let's calculate the approximation using n = 4 as an example:
Δx = (7 - 2) / 4
= 5 / 4
= 1.25
Substituting i = 1, 2, 3, 4 into x_i = 2 + (i - 1)(5 / n), we get the following values:
x_1 = 2 + (1 - 1)(5 / 4)
= 2
x_2 = 2 + (2 - 1)(5 / 4)
= 2.625
x_3 = 2 + (3 - 1)(5 / 4)
= 3.25
x_4 = 2 + (4 - 1)(5 / 4)
= 3.875
Now, we evaluate the function at the midpoints:
f(x_1) = –1(2) – 6(2^2)
= –1(2) – 6(4)
= –2 – 24
= -26
f(x_2) = –1(2.625) – 6(2.625^2)
= –1(2.625) – 6(6.890625)
= –2.625 – 41.34375
= -43.96875
f(x_3) = –1(3.25) – 6(3.25^2)
= –1(3.25) – 6(10.5625)
= –3.25 – 63.375
= -66.625
f(x_4) = –1(3.875) – 6(3.875^2)
= –1(3.875) – 6(15.015625)
= –3.875 – 90.09375
= -93.96875
Now, we can calculate the approximation of the integral:
∫(–1x–6x^2)dx ≈ Δx * Σf(x_i)
= 1.25 * (-26 + (-43.96875) + (-66.625) + (-93.96875))
= 1.25 * (-230.5625)
= -288.203125
Therefore, the approximation of the integral from 2 to 7 of (–1x–6x^2)dx using the Midpoint Rule with n = 4 is approximately -288.203125.