QJEEM21Mafaec: JEE Main 2021 Mathematics - Class 12 JEEM21Mafaec Solution: The area, enclosed by the curves and and the lines , is :

Question

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The area, enclosed by the curves

y = \sin x + \cos x

and

y = \left| {\cos x - \sin x} \right|

and the lines

x = 0,x = {\pi \over 2}

, is :

Accepted Answer

A = \int_0^{{\pi \over 2}} {\left( {(\sin x + \cos x) - \left| {\cos x - \sin x} \right|} \right)\,dx}

A = \int_0^{{\pi \over 2}} {\left( {(\sin x + \cos x) - (\cos x - \sin x)} \right)\,dx} + \int_{{\pi \over 4}}^{{\pi \over 2}} {\left( {(\sin x + \cos x) - (\sin x - \cos x)} \right)\,dx}

A = 2\int_0^{{\pi \over 2}} {\sin x\,dx + 2\int_{{\pi \over 4}}^{{\pi \over 2}} {\cos x\,dx} }

A = - 2\left( {{1 \over {\sqrt 2 }} - 1} \right) + \left( {1 - {1 \over {\sqrt 2 }}} \right)

A = 4 - 2\sqrt 2 = 2\sqrt 2 (\sqrt 2 - 1)

Option (a)

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