question_answer

The area of the region bounded by the curves \[y=|x-2|,x=1,x=3\]and the X-axis is

A) 1       

B)                        2             

C)        3             

D)        4

Correct Answer: A

Solution :

Required area \[=\int_{1}^{3}{ydx}=\int_{1}^{3}{|x-2|}dx\] \[=\int_{1}^{2}{-(x-2)dx}+\int_{2}^{3}{(x-2)}dx\] \[=\int_{1}^{2}{(2-x)dx}+\int_{2}^{3}{(x-2)}dx\] \[=\left[ 2x-\frac{{{x}^{2}}}{2} \right]_{1}^{2}+\left[ \frac{{{x}^{2}}}{2}-2x \right]_{2}^{3}\] \[=(4-2)-\left( 2-\frac{1}{2} \right)+\left( \frac{9}{2}-6 \right)-(2-4)\] \[=2-\frac{3}{2}-\frac{3}{2}+2\] \[=4-3=1\]sq unit