Calculate the Reimann sum

Question

consider the function f(x)=-(x²/3)+7

Calculate Rn which is the Reimann sum for f(x)=-(x²/3)+7 on the interval [0,4] and write the answer as a function of n without any summation sign

what is Rn?

What is the limit Rn as n approaches infinity? 

Answer 1

R_n = ∑_k=1ⁿ[-(4k/n)²/3 + 7]·4/n = 4/n[- 16n(n + 1)(2n + 1)/(18n²) + 7n] = - 32(n + 1)(2n + 1)/9n² + 28

lmR_n as n→∞∞ = - 64/9 + 28 = 188/9 = ∫₀⁴(- x²/3 + 7)dx