Chapter 8.3, Problem 2E

To determine

To rank: the following three quantities in increasing order by drawing a picture:

  ${\displaystyle \underset{1}{\overset{6}{\int }}f(x)dx},{\displaystyle \sum _{i=1}^5{a}_i},{\displaystyle \sum _{i=2}^6{a}_i}$ .

Answer to Problem 2E

  ${\displaystyle \sum _{i=2}^6{a}_i}$< ${\displaystyle \underset{1}{\overset{6}{\int }}f(x)dx}$< ${\displaystyle \sum _{i=1}^5{a}_i}$ .

Explanation of Solution

Given information: Suppose f is a continuous positive decreasing function for $x\ge 1$ and $a_n=f(n).$

Calculation:

The pictures of the following three quantities are shown below by using graphing utility.

  

The (lighter and darker) orange portion below the blue curve is ${\displaystyle \underset{1}{\overset{6}{\int }}f(x)dx}$ .The red (outline) rectangle represents ${\displaystyle \sum _{i=1}^5{a}_i}$ . The dark orange (filled-in) rectangles represents ${\displaystyle \sum _{i=2}^6{a}_i}$ ; the region with the smallest area is ${\displaystyle \sum _{i=2}^6{a}_i}$ ;the region with the largest area is ${\displaystyle \sum _{i=1}^5{a}_i}$ ;and ${\displaystyle \underset{1}{\overset{6}{\int }}f(x)dx}$ , which is the area of the orange portion below the blue curve ,is in between the two summations , thus

  ${\displaystyle \sum _{i=2}^6{a}_i}$< ${\displaystyle \underset{1}{\overset{6}{\int }}f(x)dx}$< ${\displaystyle \sum _{i=1}^5{a}_i}$ .