6. [Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule] | College Calculus: Level II | Educator.com

Midpoint and trapezoidal sums in summation notation

SOLVED:An approximation rule similar to the Trapezoid Rule is the Midpoint Rule. This rule approximates the integral ∫c^d f(x) d x by (d-c) f([d c] / 2) . Prove that if the function f:[c, d] →ℝ is continuous and its restriction f:(c, d) →ℝ has a second derivative, then for some point ζin (c, d) ∫c^d f(x) d x=(d-c) f((c d)/(2)) (1)/(24)(d-c)^3 f^''(ζ) [Hint: Let m=(c d) / 2, let t0=(d-c) / 2 and define H(t)=[∫m-t^m t f]-2 t f(m)-((t)/(t0))^3 ∫m-t0^m t0 f-2 (t^3)/(t0^2)(f) m for -t0 ≤t ≤t0 Appl

Midpoint | Definition, Theorem & Examples