Chapter 13.5, Problem 21E

a.

To determine

To approximate the area under the graph of $f\left(x\right)=x^5+2x+3\text{ on}\left[1,3\right]$ .

  $\text{use }10,20\text{ and }100\text{ rectangles}.$

Answer to Problem 21E

The area of the graph of $f\left(x\right)=x^5+2x+3$ is $R_{10}\approx 161.27,R_{20}\approx 147.97,R_{100}\approx \mathrm{137.81.}$

Explanation of Solution

Given function $f\left(x\right)=x^5+2x+3\text{ on}\left[1,3\right].$

Calculation:

  $\begin{array}{l}\text{PROGRAM:AREA}\\ \text{:Prompt N}\\ \text{:Prompt A}\\ \text{:Prompt B}\\ :\left(B-A\right)/N\to D\\ :O\to S\\ :A\to X\\ :For\left(K,1,N\right)\\ :X+D\to X\\ :S+Y_1\to S\\ :End\\ :D\ast S\to S\\ :Disp"\text{AREA IS}"\\ :\text{Disp S}\end{array}$

Use a graph utility to compute the area for various rectangles

  $\begin{array}{l}{R}_{10}\approx 161.27\\ R_{20}\approx 147.97\\ R_{100}\approx 137.81\end{array}$

Thus, the area of the graph of $f\left(x\right)=x^5+2x+3$ is $R_{10}\approx 161.27,R_{20}\approx 147.97,R_{100}\approx \mathrm{137.81.}$

b.

To determine

To approximate the area under the graph of $f$ on the given interval $100\text{ rectangles}$

  $\text{(i) }f\left(x\right)=\sin x,\text{ on}\left[0,\pi \right].\text{(ii) }f\left(x\right)=e^{-x^2}\text{, on}\left[-1,1\right].$

Answer to Problem 21E

The area of the graphs of $\text{(i) }f\left(x\right)=\sin x\text{(ii) }f\left(x\right)=e^{-x^2}$ is $2,\mathrm{1.49.}$

Explanation of Solution

Given functions are $\text{(i) }f\left(x\right)=\sin x,\text{ on}\left[0,\pi \right].\text{(ii) }f\left(x\right)=e^{-x^2}\text{, on}\left[-1,1\right].$

Calculation:

  $\begin{array}{l}\text{PROGRAM:AREA}\\ \text{:Prompt N}\\ \text{:Prompt A}\\ \text{:Prompt B}\\ :\left(B-A\right)/N\to D\\ :O\to S\\ :A\to X\\ :For\left(K,1,N\right)\\ :X+D\to X\\ :S+Y_1\to S\\ :End\\ :D\ast S\to S\\ :Disp"\text{AREA IS}"\\ :\text{Disp S}\end{array}$

The area underneath the follow curve $f\left(x\right)=\sin x,0\le x\le \pi$

Use a graph utility to compute the area for various rectangles

  $R_{100}\approx 2$

The area underneath the follow curve $f\left(x\right)=e^{-x^2},-1\le x\le 1$

Use a graph utility to compute the area for various rectangles

  $R_{100}\approx 1.49$

Thus, the area of the graph of $\text{(i) }f\left(x\right)=\sin x\text{(ii) }f\left(x\right)=e^{-x^2}$ is $2,\mathrm{1.49.}$