Chapter 11.5, Problem 27E

To determine

To calculate:

The exact area of the given function:

Answer to Problem 27E

The table of the approximate area of the region is

$n$	$4$	$8$	$20$	$50$	$100$	$\infty$
area	$40$	$38$	$36.8$	$36.32$	$36.16$	$36$

Explanation of Solution

Given information:

$n$	$4$	$8$	$20$	$50$	$100$	$\infty$
area

Function: $f\left(x\right)=2x+5$

Integral: $\left[0,4\right]$

Sum of Areas of n rectangles: $\frac{36n^2+16n}{n^2}$

Formula Used:

By using this formula:

  $A={\displaystyle \sum _{i=1}^{n}f\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)}$

  ${\displaystyle \sum _{i=1}^{n}i}=\frac{n\left(n+1\right)}{2}$

Calculation:

Consider this equation:

  $f\left(x\right)=2x+5$

By using this formula:

  $\begin{array}{l}A=\underset{x\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)}\\ A=\underset{x\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(0+\frac{\left(4-0\right)i}{n}\right)\left(\frac{4-0}{n}\right)}\\ A=\underset{x\to \infty }{\lim }{\displaystyle \sum _{i=1}^n\frac{32i}{n^2}+\frac{20}{n}}\end{array}$

By using this formula:

  ${\displaystyle \sum _{i=1}^{n}i}=\frac{n\left(n+1\right)}{2}$

Comparing with this equation:

  $\begin{array}{l}A=\frac{32}{n^2}\frac{n\left(n-1\right)}{2}+20\\ A=\frac{16n+16+20n}{n}\\ A=\frac{36n+16}{n}\end{array}$

Put the value of $n=4$ :

  $\begin{array}{l}n=4\\ A=\frac{36\times 8+16}{4}\\ A=40\end{array}$

Put the value of $n=8$ :

  $\begin{array}{l}n=8\\ A=\frac{36\times 8+16}{8}\\ A=38\end{array}$

Put the value of $n=20$ :

  $\begin{array}{l}n=20\\ A=\frac{36\times 20+16}{20}\\ A=36.8\end{array}$

Put the value of $n=50$ :

  $\begin{array}{l}n=50\\ A=\frac{36\times 50+16}{50}\\ A=36.32\end{array}$

Put the value of $n=100$ :

  $\begin{array}{l}n=100\\ A=\frac{36\times 100+16}{100}\\ A=36.16\end{array}$

Put the value of $n=\infty$ :

  $\begin{array}{l}A=\underset{x\to \infty }{\lim }\frac{36n+16}{100}\\ A=\underset{x\to \infty }{\lim }36+\frac{16}{n}\\ A=36\end{array}$

The exact area is $36$ Sq units.

$n$	$4$	$8$	$20$	$50$	$100$	$\infty$
area	$40$	$38$	$36.8$	$36.32$	$36.16$	$36$