Chapter 13.5, Problem 8E

To determine

To find : The area of the shaded region under the graph of the given function by using the indicted rectangles.

Answer to Problem 8E

Area of the shaded region is 12.9375.

Explanation of Solution

Given information:

The function of the graph is $f\left(x\right)=9x-x^3$

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:

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

The area of the shaded region under the graph using 8 rectangles having the base $\Delta x$ and the height $f\left(x_k\right)$ is

  $A=\sum _{k=1}^{8}f\left(x_k\right)\Delta x$ where $\Delta x=\frac{2-0}{8}=0.25$ , therefore,

  $\begin{array}{l}A=\sum _{k=1}^{8}f\left(x_k\right)\Delta x\\ A=\left[f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)+f\left(x_4\right)+f\left(x_5\right)+f\left(x_6\right)+f\left(x_7\right)+f\left(x_8\right)\right]\Delta x\\ A=[9\cdot \left(0.25\right)-{\left(0.25\right)}^3+9\cdot \left(0.5\right)-{\left(0.5\right)}^3+9\cdot \left(0.75\right)-{\left(0.75\right)}^3+9\cdot \left(1\right)-{\left(1\right)}^3\\ 9\cdot \left(1.25\right)-{\left(1.25\right)}^3+9\cdot \left(1.5\right)-{\left(1.5\right)}^3+9\cdot \left(1.75\right)-{\left(1.75\right)}^3+9\cdot \left(2\right)-{\left(2\right)}^3]\times 0.25\\ A=51.75\times 0.25\\ A=12.9375\end{array}$

Hence,

Area of the shaded region is 12.9375.