Question

use a midpoint Riemann sum with n=3 subintervals to approximate the area under a curve f(x)=2x +1 on the interval (0,6)

Answer 1

Delta x = (6-0)/3 = 2. Subintervals: [0,2], [2,4], [4,6]. Midpoints: 1, 3, 5.

f(1)=3, f(3)=7, f(5)=11. Midpoint Riemann sum = Δx·(3+7+11) = 2·21 = 42.

(Indeed the exact integral ∫0^6 (2x+1) dx = [x^2+x]_0^6 = 42.)