A region is enclosed by

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

y=e^{-x^{2}}

, the

x

-axis, and the line

x=2

. When this region is revolved about the vertical line

x=4

, a choice must be made to integrate with respect to

x

or with respect to

y

. Which of the following best justifies integrating with respect to

x

rather than

y

? (a) Because the axis of rotation is vertical, we must integrate with respect to

x

; integrating with

y

is impossible. (b) Integrating with respect to

y

would require inverting

e^{-x^{2}}

x=\ln(-y)

, which is impossible, so

x

is preferable. (c) With respect to

x

, both radii are simple linear functions of

x

, avoiding complicated algebra. (d) The washers method uses radii squared; integrating in

y

would force us to drop that squaring, so choosing

x

avoids a mismatch of exponents.