Chapter 5, Problem 33RE

To determine

To find: The integral by interpreting its area.

Answer to Problem 33RE

The value of the integral is: $40$ .

Explanation of Solution

Given information:

The equation is $f(x)=\left|x\right|$

Calculation:

The graph of $f(x)=\left|x\right|$ for $x\in [-4,8]$ is shown below:

  

Two triangles make up the area beneath the curve. So ${{\displaystyle \int }}_{-4}^8\left|x\right|dx$ can be found by finding the areas of these two triangles and then adding them up. The triangle on the left has a height of 4 and base of 4 so it has an area of $\frac{1}{2}(4)(4)=8$ , The triangle on the right has a height of 8 and base of 8 so it has an area of $\frac{1}{2}(8)(8)=32$ . This then gives,

  $\begin{array}{c}{{\displaystyle \int }}_{-4}^8\left|x\right|dx=8+32\\ =40\end{array}$

Therefore the required integral by interpreting its area is $40$ .