Chapter 11.2, Problem 17E

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 $f\left(x\right)=x^2$ in the interval $\left[0,4\right]$ is,

  

Given information:

  $f\left(x\right)=x^2$ in the interval $\left[0,4\right]$

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,

  $A\approx 30$

Given information:

  $f\left(x\right)=x^2$ in the interval $\left[0,4\right]$

Calculation:

The four sub-intervals are $\left[0,1\right],\left[1,2\right],\left[2,3\right],\left[3,4\right]$

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.

  $A\approx 1+4+9=14$

Subintervals	Base	Height	Area
$\left[0,1\right]$	$1$	$f\left(1\right)=1^2=1$	$\left(1\right)\left(1\right)=1$
$\left[1,2\right]$	$1$	$f\left(2\right)=2^2=4$	$\left(1\right)\left(4\right)=4$
$\left[2,3\right]$	$1$	$f\left(3\right)=3^2=9$	$\left(1\right)\left(9\right)=9$
$\left[3,4\right]$	$1$	$f\left(4\right)=4^2=16$	$\left(1\right)\left(16\right)=16$

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

  $A\approx 14$

Given information:

  $f\left(x\right)=x^2$ in the interval $\left[0,4\right]$

Calculation:

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

  $A\approx 1+4+9=14$

Subintervals	Base	Height	Area
$\left[1,2\right]$	$1$$1$	$f\left(1\right)=1^2=1$	$\left(1\right)\left(1\right)=1$
$\left[2,3\right]$	$1$	$f\left(2\right)=2^2=4$	$\left(1\right)\left(4\right)=4$
$\left[3,4\right]$	$1$	$f\left(3\right)=3^2=9$	$\left(1\right)\left(9\right)=9$

d)

To find: Average area of RRAM and.LRAM.

Average area of RRAM and.LRAM is $22$

Given information:

  $f\left(x\right)=x^2$ in the interval $\left[0,4\right]$

Calculation:

Using RRAM the area is $30$ and using LRAM the are is $14$ the average estimate of the area is

  $\frac{30+14}{2}=22$