Chapter 11.2, Problem 20E

Solution

a)

To find: Draw the graph of the function for x in the specified interval and to verify that the function is non-negative in that interval.

The graph of the function $f\left(x\right)=x^3$ in the interval $\left[0,3\right]$ is,

  

Given information:

  $f\left(x\right)=x^3$ in the interval $\left[0,3\right]$

Calculation:

  

Plotting the graph

  

b)

To find: Draw and shade the approximating rectangles for the RRAM and to find the area.

The approximating rectangles for the RRAM and the area is,

  $A\approx 36$

Given information:

  $f\left(x\right)=x^3$ in the interval $\left[0,3\right]$

Calculation:

The six sub-intervals are $\left[0,1\right],\left[1,2\right],\left[2,3\right]$

For RRAM the upper right corners of the rectangles touch the graph of the function

  

Each rectangles has a base of 1 while the heights are function value at the right-hand endpoint.

  $A\approx 1+8+27=36$

Subintervals	Base	Height	Area
$\left[0,1\right]$	$1$	$f\left(1\right)=1^3=1$	$\left(1\right)\left(1\right)=1$
$\left[1,2\right]$	$1$	$f\left(2\right)=2^3=8$	$\left(1\right)\left(8\right)=8$
$\left[2,3\right]$	$1$	$f\left(3\right)=3^3=27$	$\left(1\right)\left(27\right)=27$

c)

To find: Draw and shade the approximating rectangles for the LRAM and to find the area.

The approximating rectangles for the LRAM and the area is

  $A\approx 9$

Given information:

  $f\left(x\right)=x^3$ in the interval $\left[0,3\right]$

Calculation:

For LRAM there is no rectangle in the interval $\left[0.1\right]$

  $A\approx 1+8=9$

Subintervals	Base	Height	Area
$\left[0,1\right]$	$1$	$f\left(1\right)=1^3=1$	$\left(1\right)\left(1\right)=1$
$\left[1,2\right]$	$1$	$f\left(2\right)=2^3=8$	$\left(1\right)\left(8\right)=8$

d)

To find: Average area of RRAM and.LRAM.

Average area of RRAM and.LRAM is $22.5$

Given information:

  $f\left(x\right)=x^3$ in the interval $\left[0,3\right]$

Calculation:

Using RRAM the area is $36$ and using LRAM the are is $9$ the average estimate of the area is

  $\frac{36+9}{2}=22.5$