Engineers model the side profile of a storage tank by the curve for

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Engineers model the side profile of a storage tank by the curve

y=6\sin\left(\dfrac{\pi x}{12}\right)

for

0\le x\le12

, where

x

is measured in feet along the horizontal base and

y

is measured in feet vertically. The curve meets the base at both endpoints, enclosing a region with the

x

–axis. To fabricate the tank, this region is revolved about the horizontal line

y=-2

. Provide answers in exact form; do not evaluate any integrals. (a) Explain why the resulting cross–sections perpendicular to the

x

–axis form washers rather than solid discs. (b) Write expressions for the outer radius

R(x)

and inner radius

r(x)

in terms of

x

. (c) Write the definite integral that represents the volume of the tank generated by this revolution.