Calculate Rn for f(x)=(-x^2/3)+7 on the interval [0,4]

Question

Consider the function f(x) = (x^2 / 2) + 7 In this problem you will calculate (low: 0, up: 4) ((-x^2 / 3) + 7)*dx by using the definition (low:a, up:b) f(x)dx = lim n-->inf [ (bottom: i=1, top: n) f(xi)delta x]. The summation inside the brackets is Rn which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. Calculate Rn for f(x) = (-x^2 / 3) + 7 on the interval [0,4] and write your answer as a function of n without any summation signs. Hints: xi = 4i/n and delta x = 4/n

 

Answer 1

Rn = 4/n∑_k=1ⁿ(-16k²/n² + 7) = 4/n[ - 16n(n + 1)(2n +1)/(6n²) + 7n] = - 32(n + 1)(2n + 1)/(3n²) + 28