Chapter 11.6B, Problem 4GCE

(a)

To determine

To graph $y=\frac{1}{x}$

Answer to Problem 4GCE

  $\ln (2)=0.6931$ Square unit.

Explanation of Solution

Given:

Set $x=1$ as the lower boundary and x = 2 as the upper boundary. Record the displayed area.

Calculation:

The following is the shaded region bounded by the lines-

Area under curve $y=\frac{1}{x}$ on interval $[1,2]$ is $\ln (2)=0.6931$ square unit.

(b)

To determine

To graph $y=\frac{1}{x}$

Answer to Problem 4GCE

  $\ln (3)=1.09861$ Square unit.

Explanation of Solution

Given:

Set $x=1$ the lower boundary and x =3 as the upper boundary. Record the displayed area. Calculation:

The following is the shaded region bounded by the lines-

Area under curve $y=\frac{1}{x}$ on interval $[1,3]$ is $\ln (3)=1.09861$ square unit.

(c)

To determine

To graph $y=\frac{1}{x}$

Answer to Problem 4GCE

  $2\ln (2)=1.38269$ Square unit.

Explanation of Solution

Given:

Set $x=1$ as the lower boundary and $x=4$ as the upper boundary. Record the displayed area.

Calculation:

The following is the shaded region bounded by the lines-

Area under curve $y=\frac{1}{x}$ on interval $[1,4]$ is $2\ln (2)=1.38269$ square unit.

(d)

To determine

To use your calculator to find the values of In 2, In 3, and In 4.

Explanation of Solution

Already used in parts (a), (b) and (c).

(e)

To determine

To compare each area with each natural logarithm.

Explanation of Solution

The area in part (a) is $\ln 2$ times in part (b) is $\ln 3$ times and in part (c) $\ln 4$ times the natural logarithm.