   Chapter 11.3, Problem 2E

Chapter
Section
Textbook Problem

Suppose f is a continuous positive decreasing function for x ≥ 1 and a n = f ( n ) . By drawing a picture, rank the following three quantities in increasing order: ∫ 1 6 f ( x ) d x ∑ i = 1 5 a i ∑ i = 2 6 a i

To determine

To rank:

The given series in the increasing order

Explanation

1) Concept:

Right end point:

If the area of the rectangle a is the value of f at the right end point, then by comparing the areas of the rectangle with area under the curve y=f(x) from 1 to n,

a1+a2+a3++an1nf(x)dx

Left end point:

If the area of the rectangle a is the value of f at the left end point, then by comparing the areas of the rectangle with area under the curve y=f(x) from 1 to n,

1nf(x)dxa1+a2+a3++an-1

2) Given:

16f(x)dx ,i=15ai,i=26ai

3) Calculation:

Consider a continuous and decreasing function 16f(x)dx  whose graph is as shown below:

Consider a continuous and decreasing function i=15ai whose graph is as shown below:

The area of the rectangle a1 in the above figure is the value of f at the left end point of [1, 5], that is,

f1=a1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 