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QJEEM21Ma0d57: JEE Main 2021 Mathematics - Class 12 JEEM21Ma0d57 Solution: The graphs of sine and cosine functions, intersect each other…

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to __________.

Step-by-Step Verified Solution

JEE Main 2021 (Online) 25th February Morning Shift Mathematics - Area Under The Curves Question 95 English Explanation

A=∫π45π4(sin⁡x−cos⁡x)dxA = \int\limits_{{\pi \over 4}}^{{{5\pi } \over 4}} {(\sin x - \cos x)dx}

=[−cos⁡x−sin⁡x]π/45π/4= [ - \cos x - \sin x]_{\pi /4}^{5\pi /4}

=−[(cos⁡5π4+sin⁡5π4)−(cos⁡π4+sin⁡π4)]= - \left[ {\left( {\cos {{5\pi } \over 4} + \sin {5\pi \over 4}} \right) - \left( {\cos {\pi \over 4} + \sin {\pi \over 4}} \right)} \right]

=−[(−12−12)−(12+12)]= - \left[ {\left( { - {1 \over {\sqrt 2 }} - {1 \over {\sqrt 2 }}} \right) - \left( {{1 \over {\sqrt 2 }} + {1 \over {\sqrt 2 }}} \right)} \right]

=42=22= {4 \over {\sqrt 2 }} = 2\sqrt 2

⇒A4=(22)4=64\Rightarrow {A^4} = {\left( {2\sqrt 2 } \right)^4} = 64

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