An open trough is to be manufactured by revolving a metal plate in the –plane about…

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

Problem

An open trough is to be manufactured by revolving a metal plate in the

xy

–plane about the

x

–axis. The plate consists of two adjoining sub-regions: • Sub-region

A

: the area under

y = 1 + \dfrac{x}{2}

from

x = 0

x = 2

. • Sub-region

B

: the area under

y = 3

from

x = 2

x = 5

. The cross-sectional profile of the trough is obtained by revolving the union

A \cup B

about the

x

–axis. (a) For each sub-region, write a definite integral that represents the volume generated when that sub-region is revolved about the

x

–axis. (b) Evaluate the integral for Sub-region

A

exactly. (c) Evaluate the integral for Sub-region

B

exactly. (d) Determine the total exact volume of the trough by combining the results from parts (b) and (c).