Chapter 5.2, Problem 27E

To determine

To find : The integral.

Answer to Problem 27E

The answer: $3$

Explanation of Solution

Given information:

The integration is: ${{\displaystyle \int }}_{-1}^1(2-\left|x\right|)dx$

Calculation:

So first want to graph the function of $2-\left|x\right|$

  

This image has been zoomed in to show the area you are looking for, which is at $-1$ to 1.

Divide the area into two portions (a rectangle and a triangle, as shown in yellow), and then use the formula $b\times h$ to get the area of the rectangle. The height is 0 to 1, thus it is 1, and the base is $-1$ to 1, making it 2. Thus $(2)(1)=2$ Using $\frac{1}{2}b\times h$ , determine the triangle's area. It is 2 because the base is $-1$ to 1 and 1 because the height is $-2$ to 2. $\frac{1}{2}\times \frac{1}{2}=1$ To get 3 now, add the two areas.

Therefore the required integral of the given integration ${{\displaystyle \int }}_{-1}^1(2-\left|x\right|)dx$ is $3$ .