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  4. The value of the integral ∫-π /2π /2√1- cos 2θ dθ is equal to

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Q. The value of the integral $ \int_{-\pi /2}^{\pi /2}{\sqrt{1-{{\cos }^{2}}\theta }}\,\,d\theta $ is equal to

J & K CETJ & K CET 2012Integrals

Solution:

Let $ I=\int_{-\pi /2}^{\pi /2}{\sqrt{1-{{\cos }^{2}}\theta }\,d\theta } $
$ =\int_{-\pi /2}^{\pi /2}{|\sin \theta |\,\,d\theta } $
$ =2\int_{0}^{\pi /2}{\sin \theta \,\,\,d\theta } $
$ =2[-\cos \,\theta ]_{0}^{\pi /2} $
$ =2\left( -\cos \frac{\pi }{2}+\cos \,{{0}^{o}} \right) $
$ =2(1)=2 $