Real Analysis
1. Stemkoski and Storm - Mean Value Theorem
2. Riemann Sums

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Real Analysis

jayrold arcede, Sep 8, 2017

For teachers

1. Stemkoski and Storm - Mean Value Theorem
2. Riemann Sums

Information

1.

Stemkoski and Storm - Mean Value Theorem

– maaadmin

2.

Riemann Sums

A Riemann sum is an approximation of the form

. It is most often used to approximate the area under some function

on the closed interval

. Below are six types of sums: left-hand, midpoint, right-hand, trapezoidal, lower, and upper.

In these sums,

represents the width of each rectangle (AKA interval), defined by

. The parameter that changes depending on the type of sum is

. This determines where the function

is evaluated and thus calculates the height of each rectangle.

Left-hand:
Midpoint:
Right-hand:

A trapezoidal sum differs from the previous 3 in that

is the average of the endpoints of each interval evaulated in

Trapezoidal: where

In lower and upper sums,

where

and

such that:

Lower: is the infimum over each interval
Upper: is the supremum over each interval

, these approximations tend to the actual area between the function and x-axis which is given by:

– John Stenger

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