Question

The area between the curve $y=1-\left|x\right|$ and the x-axis is equal to

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $2$
• E. $3$

Answer 1

The correct option is A

$1$

Explanation for the correct option

Calculating the area within the straight line and the x-axis:

The function given is

$$\begin{gathered} f\left(x\right)=1-\left|x\right| \\ \Rightarrow f\left(x\right)=\left\{\begin{array}{ll}1-x& x\ge 0\\ 1+x& x<0\end{array}\right. \end{gathered}$$

Then the graph of the curves is as shown:

We see from the graph that $△AOB$ has,

height $=\mathrm{OB}=1\mathrm{units}$

base $=\mathrm{AO}=1\mathrm{units}$

We can observe that the area of the triangle can be obtained by,

$\begin{array}{rcl}Areaof∆ABC& =& 2areaof∆OAB\\ & =& 2\times \frac{1}{2}\times OB\times OA\\ & =& 1\times 1\times 1\\ & =& 1\text{sq unit}.\end{array}$

Hence, the correct answer is option (A).