That is the correct sum you should get by breaking the area into a set of rectangles with the same base, and whose height is the height of the function at the midpoint of the base.
Using the midpoint rule, find the area with two rectangles and 4 rectangles:
Problem: f(x) = x^3 btwn x = 0 and x = 1
My Process:
i) Two rectangles:
Δ x = (b-a)/n = (1-0)/2 = 1/2
A ˜ 1/2 [ f (1/4) + f (3/4) ]
A ˜ 1/2 [ (1/4)^3 + (3/4)^3 ] = . 21875
ii) Four rectangles:
Δ x = (b-a)/n = (1-0)/4 = 1/4
A ˜ 1/4 [ f (1/8) + f (3/8) + f (5/8) + f (7/8) ]
A ˜ 1/4 [ (1/8)^3 + (3/8)^3 + (5/8)^3 + (7/8)^3 ] = .2422
Is this right because when I tried doing the same process for the next problem, we got a different answer.
So how come on question # 7,
which asks f(x) = 1/x between x = 1 and x = 5
i) two rectangles:
Delta X = (b-a)/n = (5-1)/2 = 4/2 = 2
A approx. = 2 [f(1) + f(3)] = 2 [ 1 + 1/3] = 2.66666....
The way the solution manual did it was...
Delta X = (b-a)/n = (5-1)/2 = 4/2 = 2
Delta x = 1/2 x 2 + 1/4 x 2....So confused at this step???
= 2 [1/2 + 1/4]
= 2 [ 3/4]
= 3/2
What's going on?