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NDA 2019 Maths · Mathematics · Application of Integrals · medium

The area of the region enclosed by the curve y=sin x, the y-axis and the line y=1, for 0≤ x≤(π)/(2), is

Single correct option

The area of the region enclosed by the curve y=sin⁡xy=\sin x, the y-axis and the line y=1y=1, for 0≤x≤π20\le x\le\dfrac\pi2, is
  1. A. 11
  2. B. π2−1\frac\pi2-1
  3. C. π2\frac\pi2
  4. D. 1−π21-\frac\pi2
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Answer: (B) π2−1\frac\pi2-1

On [0,π2]\left[0,\frac\pi2\right] the line y=1y=1 lies above the curve y=sin⁡xy=\sin x, so the area is ∫0π/2(1−sin⁡x)dx=[x+cos⁡x]0π/2=(π2+0)−(0+1)=π2−1\int_0^{\pi/2}(1-\sin x)dx=\left[x+\cos x\right]_0^{\pi/2}=\left(\frac\pi2+0\right)-(0+1)=\frac\pi2-1. The value 11 is the area under the sine arch alone and π2\frac\pi2 is the area of the whole rectangle; the required region is the rectangle minus the arch.
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Original question written in the NDA 2019 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