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

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

When 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 expressions represents its volume? (a) π∫0π/2(cos⁡2x−sin⁡2x) dx\displaystyle \pi\int_{0}^{\pi/2}\bigl(\cos^{2}x-\sin^{2}x\bigr)\,dx (b) π∫0π/4(sin⁡2x−cos⁡2x) dx+π∫π/4π/2(cos⁡2x−sin⁡2x) dx\displaystyle \pi\int_{0}^{\pi/4}\bigl(\sin^{2}x-\cos^{2}x\bigr)\,dx+\pi\int_{\pi/4}^{\pi/2}\bigl(\cos^{2}x-\sin^{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⁡x−sin⁡x) dx\displaystyle \pi\int_{0}^{\pi/2}\bigl(\cos x-\sin x\bigr)\,dx

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