Mathematics

Problem

The region in the first quadrant bounded by

y=\sin x

and

y=\cos x

for

0\le x\le \dfrac{\pi}{2}

is revolved about the

x

-axis to form a solid. Which of the following integrals equals the volume of this solid? a)

\displaystyle \pi\int_{0}^{\pi/2}\bigl(\cos^2 x-\sin^2 x\bigr)\,dx

\displaystyle \pi\Bigl[\int_{0}^{\pi/4}\cos^2 x\,dx+\int_{\pi/4}^{\pi/2}\sin^2 x\,dx\Bigr]

\displaystyle \pi\Bigl[\int_{0}^{\pi/4}\bigl(\cos^2 x-\sin^2 x\bigr)\,dx+\int_{\pi/4}^{\pi/2}\bigl(\sin^2 x-\cos^2 x\bigr)\,dx\Bigr]

\displaystyle \pi\int_{0}^{\pi/2}\bigl(\cos x-\sin x\bigr)^2\,dx