Chapter 11.5, Problem 19E

To determine

To calculate:

The approximate area value of the following region:

  $f\left(x\right)=\frac{1}{4}{x}^3$

Answer to Problem 19E

The approximate area of the region is $1.2656$ Sq units.

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{1}{4}{x}^3$

  

Length of the interval along x-axis: $L=2$ units.

No of rectangles: $8$ rectangles

Width of the rectangle: $W=\frac{2}{8}=\frac{1}{4}$ .

Formula Used:

By using this formula:

  ${\displaystyle \sum _{i=1}^n{i}^3=\frac{n^2{\left(n+1\right)}^2}{4}}$

Calculation:

Consider this equation:

  $f\left(x\right)=\frac{1}{4}{x}^3$

The four intervals are $\left(0,\frac{1}{4}\right),\left(\frac{1}{4},\frac{1}{2}\right),\left(\frac{1}{2},\frac{3}{4}\right),\left(\frac{3}{4},1\right),\left(1,\frac{5}{4}\right),\left(\frac{5}{4},\frac{3}{2}\right),\left(\frac{3}{2},\frac{7}{4}\right),\left(\frac{7}{4},2\right)$ .

The sum of areas of the four rectangles:

  $\begin{array}{l}{\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}{\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)}\\ {\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}{\displaystyle \sum _{i=1}^{8}f\left(\frac{{\left(\frac{1}{4}i\right)}^3}{4}\right)\left(\frac{1}{4}\right)}\\ {\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}\frac{1}{1024}{\displaystyle \sum _{i=1}^8{i}^3}\end{array}$

By using this formula:

  ${\displaystyle \sum _{i=1}^n{i}^3=\frac{n^2{\left(n+1\right)}^2}{4}}$

Comparing with this formula:

  $\begin{array}{l}{\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}\frac{8^2{\left(8+1\right)}^2}{1024\times 4}\\ {\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}\frac{1}{1024}\frac{64\times 81}{4}\\ {\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}\frac{5184}{4096}\\ {\displaystyle \sum _{i=1}^{8}f\left(\frac{1}{4}i\right)\left(\frac{1}{4}\right)=}1.2656\end{array}$

The approximate area of the region is $1.2656$ Sq units.