Calculate the upper sums Un and lower sums Ln , on a regular partition of the intervals, for the following integrals.

Question

Calculate the upper sums Un and lower sums Ln, on a regular partition of the intervals, for the following integrals.

 

(a)  ∫7 1 (2−8x)dx

(b)  ∫1 0 (4+18x2)dx

(c)  ∫2 1 H(x−2)dx

where H(x) is the Heaviside function

(a) (b) answer can be an expression

Answer 1

(a) U_n = Left_n = 6/n∑_k=0^n-1(2 - 8(1 + 6k/n)) = 6/n∑_k=0^n-1(- 6 - 48k/n) = 6/n(- 6n + 6 - 24(n - 1)) = 6/n(- 30n + 30) = -180(1 - 1/n); L_n = R_n = 6/n∑_k=1ⁿ(2 - 8(1 + 6k/n) = 6/n∑_k=1ⁿ(2 - 8(1 + 6k/n)) = 6/n(- 6n - 24n - 24) =

-180(1 + 4/(5n))

(b) U_n = R_n = 1/n∑_k=1ⁿ[4 + 18(k/n)²] = 1/n(4n + 3/n(n + 1)(2n + 1)) = 10 + 9/n + 3/n²

L_n = Left_n = 1/n∑_k=0^n-1[4 + 18(k/n)²] = 1/n(4n - 4 + 3(n -1)(2n - 1)/n) = 10 - 9/n + 3/n²

(c) H(x - 2) = {0 for x < 2 and 1 for x > 2}; ∫₁²H(x - 2)dx = ∫₁²0dx = 0. SO = U_n = L_n = 0