Chapter 11.5, Problem 20E

To determine

To calculate:

The approximate area value of the following region:

  $f\left(x\right)=2-x^2$

Answer to Problem 20E

The approximate area of the region is $3.25$ Sq units.

Explanation of Solution

Given information:

  $f\left(x\right)=2-x^2$

  

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

No of rectangles: $4$ rectangles

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

Formula Used:

By using this formula:

  ${\displaystyle \sum _{i=1}^n{i}^2=\frac{n\left(n+1\right)\left(n+2\right)}{6}}$

Calculation:

Consider this equation:

  $f\left(x\right)=2-x^2$

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

The sum of areas of the four rectangles:

  $\begin{array}{l}{\displaystyle \sum _{i=-1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)=}{\displaystyle \sum _{i=-1}^{0}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)+}{\displaystyle \sum _{i=1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)}\\ {\displaystyle \sum _{i=-1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)=}{\displaystyle \sum _{i=-1}^{0}f\left(2-{\left(\frac{1}{2}i\right)}^2\right)\left(\frac{1}{2}\right)+}{\displaystyle \sum _{i=1}^{2}f\left(2-{\left(\frac{1}{2}i\right)}^2\right)\left(\frac{1}{2}\right)}\\ {\displaystyle \sum _{i=-1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)=}{\displaystyle \sum _{i=-1}^0\left(1-\frac{1}{8}{i}^2\right)+}{\displaystyle \sum _{i=1}^2\left(1-\frac{1}{8}{i}^2\right)}\end{array}$

By using this formula:

  ${\displaystyle \sum _{i=1}^n{i}^2=\frac{n\left(n+1\right)\left(n+2\right)}{6}}$

Comparing with this formula:

  $\begin{array}{l}{\displaystyle \sum _{i=-1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)=}\left(\frac{7}{8}+1\right)+\left(2-\frac{1}{8}.\frac{2\times 3\times 5}{6}\right)\\ {\displaystyle \sum _{i=-1}^{2}f\left(\frac{1}{2}i\right)\left(\frac{1}{2}\right)=}3.25\end{array}$

The approximate area of the region is $3.25$ Sq units.