Question:

The area under the curve $y = | \cos \, x - \sin \, x |, 0 \le x \le \frac{\pi}{2},$ and above x-axis is :

Updated On: Jul 7, 2022
  • $2 \sqrt{2}$
  • $2 \sqrt{2} - 2 $
  • $2 \sqrt{2} + 2$
  • 0
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The Correct Option is B

Solution and Explanation

$y = | \cos \, x - \sin x |$
Required area $= 2 \int\limits^{\pi /4}_{0} \left(\cos x - \sin x\right)dx $ $= 2\left[\sin x +\cos x\right]_{0}^{\pi/4} $ $= 2\left[\frac{2}{\sqrt{2}} - 1 \right] = \left(2\sqrt{2} -2\right) $ s units
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Concepts Used:

Area under Simple Curves

  • The area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) - given by the formula:
\[\text{Area}=\int_a^bydx=\int_a^bf(x)dx\]
  • The area of the region bounded by the curve x = φ (y), y-axis and the lines y = c, y = d - given by the formula:
\[\text{Area}=\int_c^dxdy=\int_c^d\phi(y)dy\]

Read More: Area under the curve formula