Chapter 13.5, Problem 14E

To determine

To find : The area of the region under the curve and check your answer by sketching the region and using geometry.

Answer to Problem 14E

The area of the region is 10 square units.

Explanation of Solution

Given information:

The given function is $y=2x+1$ , $1\le x\le 3$.

Calculation :

The area A of the region S that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:

  $\begin{array}{l}A=\underset{n\to \infty }{\lim }{R}_n=\underset{n\to \infty }{\lim }\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+\mathrm{......}+f\left(x_n\right)\Delta x\right]\text{ or }\\ A=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k\right)\Delta x\end{array}$

Let $f\left(x\right)=3x$ , $0\le x\le 5$.

Here,

Width:

  $\Delta x=\frac{b-a}{n}=\frac{3-1}{n}=\frac{2}{n}$

Right end point:

  $\begin{array}{l}{x}_k=a+k\Delta x\\ x_k=1+k\cdot \frac{2}{n}\end{array}$

Height:

  $\begin{array}{l}f\left(x_k\right)=2x_k+1\\ f\left(x_k\right)=2\left(1+2\cdot \frac{k}{n}\right)+1\\ f\left(x_k\right)=3+\frac{4k}{n}\end{array}$

Now, according to definition area is

  $\begin{array}{l}A=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(x_k\right)\Delta x\\ A=\underset{n\to \infty }{\lim }\sum _{k=1}^n\left[\left(3+\frac{4k}{n}\right)\cdot \frac{2}{n}\right]\\ A=\underset{n\to \infty }{\lim }\sum _{k=1}^n\left[\frac{6}{n}+\frac{8k}{n^2}\right]\\ A=\underset{n\to \infty }{\lim }\left(\sum _{k=1}^n\frac{6}{n}+\sum _{k=1}^n\frac{8k}{n^2}\right)\\ A=\underset{n\to \infty }{\lim }\left(6+\frac{8\sum _{k=1}^{n}k}{n^2}\right)\\ A=\underset{n\to \infty }{\lim }\left[6+\frac{8n\left(n+1\right)}{2n^2}\right]\\ A=\underset{n\to \infty }{\lim }\left[6+4\left(1+\frac{1}{n}\right)\right]\\ A=\underset{n\to \infty }{\lim }\left[6+4+\frac{4}{n}\right]\\ A=\left[6+4+\frac{4}{\infty }\right]\\ A=\left[6+4+0\right]\text{ (Since }\frac{a}{\infty }=0\text{)}\\ A=10\end{array}$

Now draw the graph of $y=3x$ , $0\le x\le 5$

  

The shaded region is the required region. Since it a trapezoid, therefore area is

  $\begin{array}{l}A=\frac{1}{2}\cdot \left(3+7\right)\cdot 2\\ A=\frac{20}{2}\\ A=10\end{array}$

Hence,

The area of the region is 10 square units.