# 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 #2 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

and . Over the interval  is a set of points  such that

Refinement of a partition

Norm

Upper Riemann sum

Partition

Lower Riemann sum

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

Lower Riemann sum

Upper Riemann Sum

Partition

Norm

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.