Q24 / 50 · Manipal (MET) 2022 · Mathematics

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

  • Correct answer
    A
    1 sq unit
  • B
    2 sq units
  • C
    3 sq units
  • D
    4 sq units

Solution

$\begin{aligned} \because \text { Required area } & =\int_1^3 y d x \\ & =\int_1^3|x-2| d x \\ & =\int_1^2-(x-2) d x+\int_2^3(x-2) d x \\ & =\int_1^2(2-x) d x+\int_2^3(x-2) d x \\ = & {\left[2 x-\frac{x^2}{2}\right]_1^2+\left[\frac{x^2}{2}-2 x\right]_2^3 } \\ = & {\left[4-2-\left(2-\frac{1}{2}\right)\right]+\left[\frac{9}{2}-6-(2-4)\right] } \\ = & 2-\frac{3}{2}-\frac{3}{2}+2 \\ = & 1 \text { sq unit }\end{aligned}$

Hence, the correct answer is option: 'A'.

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