Using Reimann sum definiton, find Rn

Question

Consider the function f(x)= (-x²/3)+7

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

Also calculate the limit Rn as n approaches infinity 

Answer 1

R_n = Ln = ∑_k=0^n-1(-16k²/3n²+ 7)·4/n = - 64/(3n³·∑_k=0^n-1k² + 28(n - 1)/n = - 64/(3n³)[n(n-1)(2n-1)/6 + 28(n-1)/n = 28(n - 1)/n - 32(n -1 )(2n - 1)/(9n²);

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