Chapter 5.4, Problem 50E

To determine

To find : The total area of the region between the curve and the $x$ -axis.

Answer to Problem 50E

The answer: $8$

Explanation of Solution

Given information:

The function is $y=x^3-4x,-2\le x\le 2$

Calculation:

Any area below the $x$ -axis must be counted as a positive area because   asked to find the total area rather than evaluate the integral. To determine where the function lies above and below the $x$ -axis, graph the function $f(x)=x^3-4x$ in the range $[-2,2]$ .

  

On the interval $[-2,0]$ ,

  $\begin{array}{c}{{\displaystyle \int }}_{-2}^0\left(x^3-4x\right)dx={\left.\left(\frac{1}{4}{x}^4-2x^2\right)\right|}_{-2}^0\\ =(0-0)-\left(\frac{1}{4}{(-2)}^4-2{(-2)}^2\right)\\ =0-(-4)\\ =4\end{array}$

On the interval $[0,2]$ ,

  $\begin{array}{c}{{\displaystyle \int }}_0^2\left(x^3-4x\right)dx={\left.\left(\frac{1}{4}{x}^4-2x^2\right)\right|}_0^2\\ =\left(\frac{1}{4}{(2)}^4-2{(2)}^2\right)-(0-0)\\ =-4-0\\ =-4\end{array}$

The total area:

  $\left|4\right|+|-4|=8$

Therefore, the required total area of the region between the curve and the $x$ -axis is $8$ .