Finding a Riemann sum for $f(x)=\frac{x^2}4+2$ over $[0,2]$.

Question

Consider the function f(x)= (x^2/4)+2.

Calculate Rn for f(x)= (x^2/4)+2 on the interval [0,2] and write your answer as a function of n without any summation signs.

Rn= ???

lim{n->infty} Rn= ???

i know its based on the property: ∫f(x)dx from ([a,b] =lim{n→∞} [∑ f(xi) Δx] where Δx= (a-b)/n and xi= (a+nΔx)

did i do it right?

Answer 1

Hint. One may recall the identity:

$$ \sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6, \quad n\ge1. $$

One may then consider $$ R_n=\frac2n\sum_{k=1}^n\left(\frac{k^2}{n^2}+2 \right). $$