Let be the solid obtained as follows

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

Problem

Let

S

be the solid obtained as follows. • For

0\le x\le\dfrac{\pi}{4}

, the base region is bounded by

y=\cos x

(top) and

y=\sin x

(bottom). • For

\dfrac{\pi}{4}\le x\le\dfrac{\pi}{2}

, the base region is bounded by

y=\sin x

(top) and

y=0

(bottom). Cross sections perpendicular to the

x

-axis are rectangles. On

0\le x\le\dfrac{\pi}{4}

the rectangles are squares (height equals base width); on

\dfrac{\pi}{4}\le x\le\dfrac{\pi}{2}

the rectangles have height twice the base width. Which of the following expressions equals the volume of

S

? (a)

\displaystyle \int_{0}^{\pi/4}(\cos x-\sin x)\,dx+2\int_{\pi/4}^{\pi/2}(\sin x)^{2}\,dx

(b)

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

(c)

\displaystyle \int_{0}^{\pi/4}(\cos x-\sin x)^{2}\,dx+2\int_{\pi/4}^{\pi/2}(\sin x)^{2}\,dx

(d)

\displaystyle \int_{0}^{\pi/4}(\cos x-\sin x)^{2}\,dx+\int_{\pi/4}^{\pi/2}(\sin x)^{2}\,dx