Let be the region bounded by

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

Problem

Let

R

be the region bounded by

y = \ln x + 2

, the

x

–axis,

x = 1

, and

x = 3

. When

R

is revolved about the horizontal line

y = 5

, a solid

S

is formed. (a) Express the radius of a typical circular cross–section of

S

as a function of

x

for

1 \le x \le 3

. (b) Write and evaluate a definite integral to find the exact volume of

S

. Present your answer in terms of natural logarithms.