definite integral definition

1.
dx = (5 - 1)/n = 4/n.
xi = 1 + i dx = 1 + 4i/n
f(xi) = 1 - (1 + 4i/n)/2 = 1 - 1/2 - 2i/n = 1/2 - 2i/n
f(xi) dx = (1/2 - 2i/n)(4/n) = 2/n - 8i/n^2
∫ (1 - x/2)dx = lim (n tends to infinity) Σ (2/n - 8i/n^2) where i from 1 to n.
= lim (n tends to infinity) 2/n Σ 1 - 8/n^2 Σ i
Σ 1 = n and Σ i = n(n +1)/2
So 2/n Σ 1 - 8/n^2 Σ i = (2/n)n - (8/n^2)n(n + 1)/2 = 2 - 4 - 4/n
lim (n tends to infinity) = 2 - 4 - 0 = - 2.
So ∫ ( 1 - x/2) dx from x = 1 to x = 5 is - 2.
2. Similar to Q1.
dx = [2 - (-1)]/n = 3/n
xi = -1 + i dx = - 1 + 3i/n
f(xi) = 2(-1 + 3i/n) + 1 = -1 + 6i/n
f(xi) dx = ( - 1 + 6i/n)(3/n) = - 3/n + 18i/n^2
Σ ( - 3/n + 18i/n^2) = - 3/nΣ 1 + 18/n^2 Σ i = (-3/n)n + (18/n^2)n(n + 1)/2
= - 3 + 9 + 9/n
lim (n tends to infinity) = -3 + 9 + 0 = 6
So ∫ ( 2x + 1) dx from x = -1 to x = 2 is 6.
Note : I'm not sure if this is right end point or left end point approach, but principle is the same, simply changing the value of xi.