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# Calculate the values of an integral of a step function

For define a step function on the interval by

Then, define

1. Calculate .
2. Find all values of such that .

1. We calculuate:

2. .

Note: There is an error in the book. The answers in the back of the book claim that , which is incorrect.

# 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:

# Prove an identity for the sum of integrals of particular step functions

Prove

Proof. From this exercise (1.11 #4, part b) we know

But, since is constant on the open subintervals of the partition

which contains every integer between and , we have on the open subintervals of (since there are no integers in the open subintervals). Hence, . Thus,

# Find a step function whose integral has specific properties

Find a step function such that

Let

Then,

as requested.

# Compute the integrals of some step functions

Compute the following integrals where is the greatest integer less than or equal to .

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

1. .
2. .
3. .
4. .
5. . (See, 1.11 #4 (d) here).
6. .

# Every step function is a linear combination of characteristic functions

We define a characteristic function, , on a set of points on by

Let be a step functions taking the (constant) value on the th open subinterval of a partition of . Prove that for each , we have

Proof. First, we note that the open subintervals of some partition of are necessarily disjoint since . Hence, if then for exactly one .
So, we have

for all , and for any . Further, by definition of , we know if . So,

for each

# Draw the graphs of even more step functions

Draw the graph of each function defined below.

1. for .
2. for .
3. for .
4. for .

1. The graph is:

2. The graph is:

3. The graph is:

4. The graph is: