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  4. Area enclosed between the curves y= sin 2 x and y= cos 2 x in the interval 0 ≤ x ≤ π, is equal to

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Q. Area enclosed between the curves $y=\sin ^2 x$ and $y=\cos ^2 x$ in the interval $0 \leq x \leq \pi$, is equal to

Application of Integrals

Solution:

image
Area $=\int\limits_{\pi / 4}^{3 \pi / 4}\left(\sin ^2 x-\cos ^2 x\right) d x$
$\left.=-\int\limits_{\pi / 4}^{3 \pi / 4} \cos 2 x d x=\frac{-1}{2}(\sin 2 x)\right]_{\pi / 4}^{3 \pi / 4} $
$=\frac{-1}{2}(-1-1)=1 $