To find: The relation between area, lower sum, and upper sum for an increasing function f.
The relation between area, lower sum, and upper sum for a function is .
The curve f is an increasing continuous function.
Describe the relation between , A, and as below:
Therefore, the terms are related as .
To prove: The relation .
The relation is proved.
The expression for upper estimate of area is shown below:
Here, the right endpoint height of the first rectangle is , the width of the interval is , the right endpoint height of the second rectangle is , and the right endpoint height of nth rectangle is .
The expression for lower estimate of area is shown below:
Here, the left endpoint height of the first rectangle is , the left endpoint height of the second rectangle is , and the left endpoint height of nth rectangle is .
Subtract Equation (2) from Equation (1) as shown below:
The width of the interval is expressed as shown below:
Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.
Substitute for , a for , and for in Equation (3).
Hence, the relation is proved.
Draw the diagram for an increasing function f with n rectangles as shown in Figure (1).
Refer to Figure (1).
Therefore, the estimate is reassembled and forms a single rectangle with area of .
To prove: is less than .
Yes, the expression is proved.
Refer to part (a).
The value of is less than the area A.
Therefore, it can be expressed as . Also it is valid for the inequality .
Substitute for .
Therefore, the expression is valid and it is proved.
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