Consider the region in the –plane bounded above by the curve

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the region

R

in the

xy

–plane bounded above by the curve

y=e^{-x/2}

, below by the

x

–axis, and between

x=0

and

x=3

. The region

R

is revolved about the horizontal line

y=5

to produce a solid. (a) Derive a definite integral expression that represents the volume of the solid. (b) Evaluate the integral found in part (a) to obtain the exact volume. (c) Let

R_{L}

denote the analogous region bounded by

y=e^{-x/2}

, the

x

–axis, and

x=0

x=L

, where

L>0

. The region

R_{L}

is revolved about

y=5

to generate a solid of volume

V(L)

. Show that

\displaystyle \lim_{L\to\infty} V(L)

exists and determine its exact value. (d) Explain, in the context of geometric convergence, why the limit in part (c) is finite even though the base region

R_{L}

grows without bound as

L\to\infty

. Reference specific features of the integrand in your explanation.

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