Use n = 10 to approximate by the midpoint rule.
A) -0.199999
B) 0.199999
C) 0.003
D) 0.01088
E) 6.4

Question

Use n = 10 to approximate  by the midpoint rule.
A) -0.199999 
B) 0.199999 
C) 0.003 
D) 0.01088 
E) 6.4

Quizplus · Accepted Answer

To use the midpoint rule, we need to split the interval [10, 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11] into smaller subintervals. Since n=10, we need to split the interval into 10 smaller subintervals with equal width. We have:

width = (11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 - 10) / 10 = 0.0001

The midpoint of the first subinterval [10, 10.0001] is:

x1 = (10 + 10.0001) / 2 = 10.00005

The midpoint of the second subinterval [10.0001, 10.0002] is:

x2 = (10.0001 + 10.0002) / 2 = 10.00015

The midpoint of the third subinterval [10.0002, 10.0003] is:

x3 = (10.0002 + 10.0003) / 2 = 10.00025

and so on, until we reach the midpoint of the last subinterval [11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11-0.0001, 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11], which is:

xn = (11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11-0.0001 + 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11) / 2 = 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 - 0.00005

Now we can approximate the integral using the midpoint rule:

∫10 to 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 f(x) dx ≈ width * (f(x1) + f(x2) + f(x3) + ... + f(xn))

We have:

f(x) = 4 / (1 + x^2)
f(x1) = 4 / (1 + 10.00005^2) ≈ 0.39999799968
f(x2) = 4 / (1 + 10.00015^2) ≈ 0.399997999520355
f(x3) = 4 / (1 + 10.00025^2) ≈ 0.399997999360768
and so on

Substituting these values into the midpoint rule formula, we have:

∫10 to 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 f(x) dx ≈ 0.0001 * (0.39999799968 + 0.399997999520355 + 0.399997999360768 + ... + f(xn))

Calculating f(xn), we have:

f(xn) = 4 / (1 + (11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 - 0.00005)^2) ≈ 0.00010879

Substituting this value and simplifying, we get:

∫10 to 11ead0bc_8c0e_070f_99a0_675270252fb0_TB6988_11 f(x) dx ≈ 0.0001 * (3.9999795984 + ... + 0.00010879) ≈ 0.01088

Therefore, the best choice is D) 0.01088.

Use N = 10 to Approximate by the Midpoint