Use the Midpoint Rule with n = 4 to approximate the area of the following region.
A) 3
B) 8
C) 2
D) 1
E) 6

Question

Use the Midpoint Rule with n = 4 to approximate the area of the following region. ​  ​  ​
A) 3 
B) 8 
C) 2 
D) 1 
E) 6

Quizplus · Accepted Answer

Using the Midpoint Rule with n = 4, we will divide the interval [0,2] into 4 subintervals of equal width:

Δx = (2-0)/4 = 0.5

We will evaluate the function at the midpoint of each subinterval, and multiply by the width of the subinterval:

f(0.25)(0.5) + f(0.75)(0.5) + f(1.25)(0.5) + f(1.75)(0.5)

Using a calculator, we can evaluate:

f(0.25) = 2
f(0.75) = 2.25
f(1.25) = 2.5
f(1.75) = 2.25

Therefore, the approximation to the area is:

2(0.5) + 2.25(0.5) + 2.5(0.5) + 2.25(0.5) = 2.625

The closest choice is C: 2.

Use the Midpoint Rule with N = 4 to Approximate