Chapter 13.5, Problem 10E

a.

To determine

To find the area of the graph of $f\left(x\right)=25-x^2,x\in \left[0,5\right]$ .

Answer to Problem 10E

The area of the graph of $f\left(x\right)=25-x^2$ is $70.$

Explanation of Solution

Given function: $f\left(x\right)=25-x^2,x\in \left[0,5\right]$

Calculation:

Let, $f\left(x\right)=25-x^2,x\in \left[0,5\right]$

  $\begin{array}{l}\Delta x=\frac{b-a}{n}\\ \Delta x=\frac{5-0}{5}\\ =1\end{array}$

Graph and the rectangles as shown below:

  

The value $A_1$ underestimates the area as all rectangles are under the function’s graph.

Compute the area: Use the right endpoints.

  $\begin{array}{l}{A}_1={\displaystyle {\sum }_{K=1}^{5}f}\left(x_k\right)\Delta x\\ =\left[f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)+f\left(x_4\right)+f\left(x_5\right)\right]\Delta x\\ =\left(25-1^2+25-2^2+25-3^2+25-4^2+25-5^2\right)\cdot 1\\ =70\end{array}$

Thus, the area of the graph of $f\left(x\right)=25-x^2$ is $70.$

b.

To determine

To find the area of the graph of $f\left(x\right)=25-x^2,x\in \left[0,5\right]$

Answer to Problem 10E

The area of the graph of $f\left(x\right)=25-x^2$ is $95.$

Explanation of Solution

Given function $f\left(x\right)=25-x^2,x\in \left[0,5\right]$

Calculation:

Graph and the rectangles as shown below:

  

The value $A_2$ overestimates the area as all rectangles have a region above the function’s graph.

Use the left endpoints.

  $\begin{array}{l}{A}_2={\displaystyle {\sum }_{K=1}^{5}f}\left(x_k\right)\Delta x\\ =\left[f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)+f\left(x_4\right)+f\left(x_5\right)\right]\Delta x\\ =\left(25-0^2+25-1^2+25-2^2+25-4^2+25-4^2\right)\cdot 1\\ =95\end{array}$

Thus, the area of the graph of $f\left(x\right)=\frac{1}{x}$ is $95.$