Mathematics

Problem

A solid has square cross sections perpendicular to the

x

-axis. The base of the solid is the region between the curves

y=x^{2}

and

y=6-x^{2}

on the interval

-2\le x\le 2

. Four students propose the following integral expressions for the volume of the solid. I.

\displaystyle \int_{-2}^{2}\bigl((6-x^{2})-x^{2}\bigr)\,dx

II.

\displaystyle \int_{-2}^{2}\bigl((6-x^{2})-x^{2}\bigr)^{2}\,dx

III.

\displaystyle \int_{-2}^{2}\bigl((6-x^{2})+x^{2}\bigr)^{2}\,dx

IV.

\displaystyle \int_{-2}^{2}\Bigl((6-x^{2})^{2}-(x^{2})^{2}\Bigr)\,dx

Which integral correctly represents the volume? a)

I\text{ only}

III\text{ only}

IV\text{ only}

II\text{ only}