Mathematics

Problem

For a solid whose base is the region bounded by the graphs of

y=0

and

y=\sin x

0\le x\le \pi

, the cross-sections perpendicular to the x-axis are rectangles. At each

x

, the rectangle’s height equals twice the square of its base width. Which of the following definite integrals represents the volume of the solid? a)

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

\displaystyle \int_{0}^{\pi} 2(\sin x)^3\,dx

\displaystyle \int_{0}^{\pi} 2\sin x\,dx

\displaystyle \int_{0}^{\pi} (\sin x)^3\,dx