Chapter 6, Problem 31RE

To determine

To evaluate: the integral by interpreting its area.

Answer to Problem 31RE

Hence, the answer is: ${{\displaystyle \int }}_{-4}^8\left|x\right|dx=40$

Explanation of Solution

Given information:

Given expression is : ${{\displaystyle \int }}_{-4}^8\left|x\right|dx$

Calculation:

If $y=f(x)$ is nonnegative and capable of integration over closed interval $[a,b]$ then the area under the curve $y=f(x)$ from $a$ to $b$ is the integral of $f$ from $a$ to $b$ , The given integral is

  ${{\displaystyle \int }}_{-4}^8\left|x\right|dx$

Take the given integrand as $y=\left|x\right|$

The graph of the function between the limits -4to 8 is shown below:

  

So,

  ${{\displaystyle \int }}_{-4}^8\left|x\right|dx=($ area of triangle of left side of $y$ -axis) $+($ area of triangle of right side of $y$ -axis)

  $\begin{array}{l}=\frac{1}{2}\left(4\right)\left(4\right)+\frac{1}{2}\left(8\right)\left(8\right)\\ =8+32\\ =40\end{array}$

Hence, the answer is: ${{\displaystyle \int }}_{-4}^8\left|x\right|dx=40$