／var／logmarcus chiu

Riemann／Riemannian Sums (Right／Left Endpoint - Midpoint - Trapezoidal Rule) - Riemann Integral

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Riemann/Riemannian Sums - Riemann Integral
  • a Riemann sum is a certain kind of approximation of an integral by a finite sum
  • a Riemann sum 𝑆 of 𝑓 over interval [𝑎,𝑏] is defined as:
  • where:
    • 𝛥𝑥𝑖 = 𝑥𝑖 - 𝑥𝑖-1
    • 𝑥̃𝑖 ∊ [𝑥𝑖-1, 𝑥𝑖]
  • a Riemann integral is the limit of the Riemann sums of a function as the partitions get finer

Riemann Sums - Types

Type

Description

Left Riemann Sum
  • 𝑥̃𝑖 = 𝑥𝑖-1

Right Riemann Sum
  • 𝑥̃𝑖 = 𝑥𝑖

Middle Riemann Sum
  • 𝑥̃𝑖 = (𝑥𝑖 + 𝑥𝑖-1) / 2

Upper Riemann Sum
  • 𝑥̃𝑖 = 𝑠𝑢𝑝 𝑓([𝑥𝑖-1, 𝑥𝑖])

Lower Riemann Sum
  • 𝑥̃𝑖 = 𝑖𝑛𝑓 𝑓([𝑥𝑖-1, 𝑥𝑖])

Trapezoidal Reimann Sum
  • taking the average of both:
    • left Riemann sum
    • right Riemann sum