Introduction to Analysis : Riemann Integral, Riemann Sums, & Improper Riemann Integration

Example Questions

Example Question #1 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?   ,  and    ,  and be bounded   ,  and be bounded   ,  and    ,  and be bounded   ,  and be bounded

Explanation:

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where represents the upper integral and the following are defined:

1. The upper integral of on is where is a partition of .

2. The lower integral of on is where is a partition of .

3. If 1 and 2 are the same then the integral is said to be if and only if   ,  and be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if   ,  and be bounded.

Example Question #1 : Intro Analysis

What term has the following definition.  and . Over the interval is a set of points such that Lower Riemann sum

Refinement of a partition

Upper Riemann sum

Norm

Partition

Partition

Explanation:

By definition

If  and .

A partition over the interval is a set of points such that .

Therefore, the term that describes this statement is partition.

Example Question #3 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

The __________ of a partition is Norm

Upper Riemann Sum

Lower Riemann sum

Partition

Refinement of a partition

Norm

Explanation:

By definition

If  and .

A partition over the interval is a set of points such that .

Furthermore,

The norm of the partition is Therefore, the term that describes this statement is norm. 