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NDA 2020 Maths · Mathematics · Definite Integrals · medium

The modulus integral ∫-2^3|x-1| dx, which measures the area between the graph and the x-axis, equals

Single correct option

The modulus integral ∫−23∣x−1∣ dx\displaystyle\int_{-2}^{3}|x-1|\,dx, which measures the area between the graph and the x-axis, equals
  1. A. 52\dfrac52
  2. B. 92\dfrac92
  3. C. 132\dfrac{13}{2}
  4. D. 72\dfrac{7}{2}
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Answer: (C) 132\dfrac{13}{2}

Split at x=1x=1 where the expression changes sign: ∫−21(1−x) dx+∫13(x−1) dx=322+222=92+2=132\displaystyle\int_{-2}^1(1-x)\,dx+\int_1^3(x-1)\,dx=\dfrac{3^2}{2}+\dfrac{2^2}{2}=\dfrac92+2=\dfrac{13}2. Without the modulus, ∫−23(x−1)dx=52\displaystyle\int_{-2}^3(x-1)dx=\dfrac52, which is the first distractor. The value 92\dfrac92 is only the left portion. Interpret the integral as the area of two right triangles with legs 3,33,3 and 2,22,2.
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