Integration

Δx = (4 - 0)/n = 4/n.
xi = 4i/n for right - hand endpoints.
f(xi) = ( - xi^2/3 + 5) = - 16i^2/3n^2 + 5
So Σ f(xi)Δx = Σ (4/n) f(xi) = Σ [- 64i^2/3n^3 + 20/n]
= Σ [ - 64i^2/3n^3] + Σ [ 20/n]
= - 64/3n^3 Σ i^2 + 20/n Σ 1
= (- 64/3n^3)[ 2n^3 + 3n^2 + n]/6 + (20/n)(n)
= - 64/9 - 32/3n - 32/9n + 20
= - 64/9 + 20 when n tends to infinity.
= 116/9.
So the integral = 116/9.

2013-12-06 08:10:36 補充：
Remark : Summation inside the bracket,Rn, is therefore = - 64/9 - 32/3n - 32/9n + 20.