Chapter 14, Problem 43RE

(a)

To determine

To find: The integral represent.

Answer to Problem 43RE

The integral represent is the area under the function $f\left(x\right)=9-x^2$ .

Explanation of Solution

Given information:

The given function is ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .

Calculation:

Consider the integral ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ ,

The integral ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ represents the area under the function $f\left(x\right)=9-x^2$ from $1to3$ .

(b)

To determine

To graph: The integral of ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .

Explanation of Solution

Given information:

The given function is ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .

Graph:

Consider the function ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .

The graph for the given function ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .is shown in figurer-(1)

  

Figure-(1)

Interpretation: Graph for the functions ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ and shade the area the function from $1to3$ shows in figure (1).

(c)

To determine

To find: The area of integral.

Answer to Problem 43RE

The integral represent is the area is ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}=\frac{80}{3}$ .

Explanation of Solution

Given information:

The given function is ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ .

Calculation:

Calculate the area,

Consider the integral ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$ using a graphing utility.

On a graphing utility enter the function ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}$

Therefore, the area of function is ${\displaystyle {\int }_{-1}^3\left(9-x^2\right)dx}=\frac{80}{3}$ .