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# Prove an inequality relating sums and integrals

Prove that

where is a nonnegative, increasing function defined for all .

Use this to deduce the inequalities

by taking .

Proof. Since is increasing we know (where denotes the greatest integer less than or equal to ) for all . Define step functions and by

Then and are constant on the open subintervals of the partition

So,

Since is integrable we must have

for every pair of step functions . Hence,

Next, if we take (which is nonnegative and increasing on ) we have

From this we then get two inequalities

Therefore,

# Prove some statements about integrals of bounded monotonic increasing functions

Consider a bounded, monotonic, real-valued function on the interval . The define sequences

1. Prove that

and that

2. Prove that the two sequences and converge to .
3. State and prove a generalization of the above to interval .

1. Proof.First, we define two step functions,

where denotes the greatest integer less than or equal to . Then we define a partition of ,

For any we have

So, and are constant on the open subintervals of the partition .
Since is monotonically increasing and for all (by the definition of and ) we have

2. Proof. From part (a) we have

since . Since does not depend on we have

Therefore,

3. Claim: If is a real-valued function that is monotonic increasing and bounded on the interval , then

for and defined as follows:

Proof. Let

be a partition of the interval . Then, define step functions and with and for . By these definitions we have for all (since is monotonic increasing). Since is bounded and monotonic increasing it is integrable, and

And, since , and we have

# Prove inequalities of the logarithm with respect to some series

Consider a partition of the interval for some .

1. Find step functions that are constant on the open subintervals of and integrate to derive the inequalities:

2. Give a geometric interpretation of the inequalities in part (a).
3. Find a particular partition (i.e., choose particular values for ) to establish the following inequalities for ,

1. Proof. We define step function and by

Since is strictly decreasing on , we have

Therefore, using the definition of the integral of a step function as a sum,

2. Geometrically, these inequalities say that the area under the curve lies between the step functions that take on the values and for each .
3. Proof. To establish these inequalities we pick the partition,

Then, applying part (a) we have

The final line follows since so the sum on the left starts with and the sum on the right only runs to . These were the inequalities requested

# Prove the additive property of integrals of step functions

For step functions defined on an interval , prove

Proof. Let be a partition of such that is constant on the open subintervals of (we know such a exists since we can take the common refinement of the partition of on which and individually are constant on open subintervals). Let if for . Then, working from the definition of the integral of a step function on an interval, we have

# More integrals of step functions

1. Compute

2. For prove

1. We compute,

2. Proof. Let . Then is a partition of and is constant on the open subintervals of . Further, for we have . Thus, we have (using this exercise and this exercise to evaluate some of the sums),

# Find some formulas for the integral of the step function [t]^2

1. For , prove

2. For , with , define

Draw the graph of on the interval .

3. Find all real such that

1. Proof. Let . Then is a partition of and is constant on the open subintervals of . Further, for . So,

The second to last line follows from this exercise (I.4.7, #6)

2. The graph is:

3. By inspection, we have, .

# Compute integrals of some more step functions

1. Prove

2. Compute

1. Proof. Let be a partition of . Then is constant on the open subintervals of , so,

2. We compute,

# Compute some integrals of step functions

1. Let , prove

2. Let , and define

Draw the graph of for .

1. Proof. Let be a partition of . Then, by the definition of the greatest integer function, is constant on the open subintervals of , so

The final equality follows from here (I.4.7, #5)

2. The graph is:

# Draw the graphs of some functions

Draw the graphs of the functions defined below on the interval , and if it is a step function find a partition such that the function is constant on the open subintervals of .

1. .
2. .
3. .
4. .
5. .
6. .

1. This is not a step function. The graph is below.

2. This is not a step function. The graph is below.

3. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

4. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

5. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.

6. This is a step function and it is constant on the open subintervals of the partition, . The graph is below.