Chapter 4: Q 10 (page 339)

Consider the area between the graph of a positive function
f and the x-axis on an interval [a, b]. Explain why the
upper sum approximation for this area with n = 8 boxes
must be smaller than or equal to the upper sum approximation
with n = 4 boxes. It may help to sketch some
examples.

Short Answer

Expert verified

To explain, the upper sum approximation with $n = 8$ boxes for this region must be smaller or equal to the upper sum approximation with $n = 4$ boxes.

Step by step solution

Solution Explanation

On the interval $[a, b]$ , consider the area between the graph of a positive function $f$ and the $x -$ axis.

Step 2:

In two separate scenarios, the number of rectangles taken is $n = 8$ .

The top total is always bigger than or equal to the signed area in question.

When, the number of sub intervals that is the number of rectangles increases, the width of the subintervals reduces which contributes to a lower sum.

As the number of sub intervals, or rectangles, reduces, the width of the sub intervals grows, resulting in a larger sum.

As a result, the area upper approximation with $n = 8$ , $n = 4$ must be smaller or equal to the upper sum approximation with $n = 8$ .

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Consider the area between the graph of a positive function
f and the x-axis on an interval [a, b]. Explain why the
upper sum approximation for this area with n = 8 boxes
must be smaller than or equal to the upper sum approximation
with n = 4 boxes. It may help to sketch some
examples.

Short Answer

Step by step solution

Solution Explanation

Step 2:

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Consider the area between the graph of a positive functionf and the x-axis on an interval [a, b]. Explain why theupper sum approximation for this area with n = 8 boxesmust be smaller than or equal to the upper sum approximationwith n = 4 boxes. It may help to sketch someexamples.

Short Answer

Step by step solution

Solution Explanation

Step 2:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Probability and Statistics

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Consider the area between the graph of a positive function
f and the x-axis on an interval [a, b]. Explain why the
upper sum approximation for this area with n = 8 boxes
must be smaller than or equal to the upper sum approximation
with n = 4 boxes. It may help to sketch some
examples.