Chapter 8.2, Problem 44E

To determine

To Find:

To find the area between the curves $y=f\left(x\right)$ and $y=2f\left(x\right)$ from $x=a$ to $x=b$ .

Answer to Problem 44E

The area between the curves is $4sq.units$ .

Explanation of Solution

Given information:

The given area of the region between the graph of a positive continuous function $f$ and the x-axis from $x=a$ to $x=b$ is $4sq.units$ .

The area between the graph of two functions can be calculated through an integral functions.

That is, the definite integrals of the absolute value of the difference of the functions are given by,

  $A={\displaystyle \underset{a}{\overset{b}{\int }}\left|f\left(x\right)-g\left(x\right)\right|}dx$

Now we have to find the area between the curves $y=f\left(x\right)$ and $y=2f\left(x\right)$ from $x=a$ to $x=b$ .

  $\begin{array}{l}A={\displaystyle \underset{a}{\overset{b}{\int }}\left|2f\left(x\right)-f\left(x\right)\right|}dx\\ A={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx\end{array}$

From given we know that the area enclosed by the function $y=f\left(x\right)$ and $y=2f\left(x\right)$ from $x=a$ to $x=b$ is just same as the area enclosed by $y=f\left(x\right)$ and the x-axis between $x=a$ to $x=b$ , since the function is positive over this range.

Therefore, The area between the curves is $4sq.units$ .