Question

The area of the region bounded by the curves $\mathrm{y}=|\mathrm{x}-2|,\mathrm{x}=1,\mathrm{x}=3$ and the x-axis is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A

$1$

Explanation for correct option

Calculating the area of the region bounded by the curves

Given, the curves are $\mathrm{y}=|\mathrm{x}-2|,\mathrm{x}=1,\mathrm{x}=3$

Visualizing the given equations on a graph:

The limit of the area will be:

$$\begin{gathered} x=1\text{to}x=2,y=x-x \\ x=2\text{to}x=3,y=x-2 \end{gathered}$$

So,

$$\begin{gathered} {\int }_1^3|\mathrm{x}-2|={\int }_1^2(2-\mathrm{x})\mathrm{dx}+{\int }_2^3(\mathrm{x}-2)\mathrm{dx} \\ ={\left[2\mathrm{x}-\frac{{\mathrm{x}}^2}{2}\right]}_1^2+{\left[\frac{{\mathrm{x}}^2}{2}-2\mathrm{x}\right]}_2^3 \\ =\left[\left(4-\frac{4}{2}\right)-\left(2-\frac{1}{2}\right)\right]+\left[\left(\frac{9}{2}-6\right)-\left(\frac{4}{2}-4\right)\right] \\ =1\mathrm{sq}\mathrm{unit} \end{gathered}$$

Hence, the correct answer is Option (A).