The area of the region enclosed by the curve f ( x ) = max { sin x , cos x } , x and the x -axis is [2023]

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Solution

Q.
The area of the region enclosed by the curve $f (x) = \max {\sin x, \cos x}$ , $- π \leq x \leq π$ and the $x$ -axis is [2023]

1 $4 (\sqrt{2})$

2 $2 (\sqrt{2} + 1)$

3 $2 \sqrt{2} (\sqrt{2} + 1)$

4 $4$

Ans.
(4)

We have $f (x) = \max {\sin x, \cos x}, - π \leq x \leq π$

Now, the area of the region enclosed by the given curve and the $x$ –axis is

$=$ $| \int_{- π}^{- 3 π / 4} \sin x d x |$ $+$ $| \int_{- 3 π / 4}^{- π / 2} \cos x d x |$ $+$ $| \int_{- π / 2}^{π / 4} \cos x d x |$ $+$ $| \int_{π / 4}^{π} \sin x d x |$

$= \frac{- 1}{\sqrt{2}} + 1 + 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} + 1 + 1 + \frac{1}{\sqrt{2}} = 4$

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