The region bounded by and for is revolved about the -axis

MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

The region bounded by y=sin⁡xy=\sin x and y=cos⁡xy=\cos x for 0≤x≤π20\le x\le \dfrac{\pi}{2} is revolved about the xx-axis. Which of the following definite integrals equals the volume of the solid generated? (a) π∫0π/2(cos⁡x−sin⁡x)2 dx\displaystyle \pi\int_{0}^{\pi/2}(\cos x-\sin x)^{2}\,dx (b) π∫0π/2(sin⁡2x+cos⁡2x) dx\displaystyle \pi\int_{0}^{\pi/2}\bigl(\sin^{2}x+\cos^{2}x\bigr)\,dx (c) π∫0π/4(cos⁡2x−sin⁡2x) dx+π∫π/4π/2(sin⁡2x−cos⁡2x) dx\displaystyle \pi\int_{0}^{\pi/4}\bigl(\cos^{2}x-\sin^{2}x\bigr)\,dx+\pi\int_{\pi/4}^{\pi/2}\bigl(\sin^{2}x-\cos^{2}x\bigr)\,dx (d) π∫0π/2(cos⁡2x−sin⁡2x) dx\displaystyle \pi\int_{0}^{\pi/2}\bigl(\cos^{2}x-\sin^{2}x\bigr)\,dx

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