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NDA 2020 Maths · Mathematics · Application of Integrals · medium

The area of the region bounded by the curves y=|x| and y=6-x^2 is

Single correct option

The area of the region bounded by the curves y=∣x∣y=|x| and y=6−x2y=6-x^2 is
  1. A. 223\dfrac{22}{3}
  2. B. 443\dfrac{44}{3}
  3. C. 403\dfrac{40}{3}
  4. D. 1414
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Answer: (B) 443\dfrac{44}{3}

For x≥0x\ge0 the curves meet where x=6−x2x=6-x^2, i.e. x=2x=2; by symmetry they also meet at x=−2x=-2. The parabola lies above y=∣x∣y=|x| in between, so area =2∫02(6−x2−x) dx=2[6x−x33−x22]02=2(12−83−2)=443=2\displaystyle\int_0^2(6-x^2-x)\,dx=2\left[6x-\dfrac{x^3}3-\dfrac{x^2}2\right]_0^2=2\left(12-\dfrac83-2\right)=\dfrac{44}3. The value 223\dfrac{22}3 is only the right half of the region, found by forgetting the symmetry factor 22.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake