Consider positive constants and

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

Problem

Consider positive constants

a>0

and

c>0

. Let

R

be the region bounded by the curve

y = e^{-x}

, the

x

–axis,

x=0

, and

x=a

. Revolving

R

about the horizontal line

y = -c

produces a solid

S

. (a) Derive a definite integral expression for the volume

V(a,c)

of the solid

S

. (b) Evaluate the integral from part (a) to obtain an explicit formula for

V(a,c)

in terms of

a

and

c

. (c) Using

a = \ln 4

and

c = 2

, calculate the exact volume of the corresponding solid. (d) Determine

\displaystyle \lim_{a\to\infty} V(a,c)

in terms of

c

, and interpret this limit in the context of the solid’s geometry.