Mathematics

Problem

A solid has as its base the region in the first quadrant bounded by the

x

-axis and the curve

y=\sin x

for

0\le x\le \pi

. Cross sections perpendicular to the

x

-axis alternate in shape: • For

0\le x\le \tfrac{\pi}{3}

, each cross section is a square. • For

\tfrac{\pi}{3}<x\le \pi

, each cross section is a rectangle whose height equals the side length of the square at the same

x

and whose width is twice that height. What is the exact volume of the solid? a)

\dfrac{2\pi}{3}+\dfrac{\sqrt{3}}{4}

\dfrac{5\pi}{6}+\dfrac{\sqrt{3}}{8}

\dfrac{5\pi}{6}-\dfrac{\sqrt{3}}{8}

\dfrac{\pi}{2}+\dfrac{\sqrt{3}}{4}