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Mathematics Question on applications of integrals

Find area bounded by the curves yy = maxsinx,cosxmax\\{sin x, cos x\\} and x−axisx-axis between x=−πx = - \pi and x=πx = \pi

A

2+22 + \sqrt{2}

B

2\sqrt{2}

C

1+21 + \sqrt{2}

D

222\sqrt{2}

Answer

222\sqrt{2}

Explanation

Solution

area bounded by the curves

∫−π−3π4sinx dx+∫−3π4π4cosx dx+∫π4πsinx dx\int_{-\pi}^{-3\frac{\pi}{4}}sinx\,dx+\int_{-3\frac{\pi}{4}}^{\frac{\pi}{4}}cosx\,dx+\int_{\frac{\pi}{4}}^{\pi}sinx\,dx

−cos x∣−π−3π4+sin x∣−3π4π4+ −cos x∣π4π-cos\,x|^{-3\frac{\pi}{4}}_{-\pi}+sin\,x|^{\frac{\pi}{4}}_{-3\frac{\pi}{4}}+ \,-cos\,x|^{\pi}_{\frac{\pi}{4}}

=(12−1)+(12+12)+(1+12)=(\frac{1}{\sqrt2}-1)+(\frac{1}{\sqrt2}+\frac{1}{\sqrt2})+(1+\frac{1}{\sqrt2})

=22=2\sqrt2

The correct option is (D): 222\sqrt{2}