Mathematics

Problem

The parabolic arch of a bridge is modeled by

y=4-x^{2}

on the interval

-2\le x\le2

. The region between this arch and the

x

–axis is used to form a decorative cylindrical tunnel by revolving the region about the vertical line

x=5

. (a) Write an integral with respect to

x

that gives the exact volume of the tunnel. (b) Express

x

as a function of

y

and determine the corresponding

y

–limits for the region. (c) Using your expression from part (b), write an integral with respect to

y

that also computes the volume. (d) Evaluate whichever of the two integrals you consider simpler to obtain the exact volume in cubic units.