For a positive parameter

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

Problem

k>0

, let R(k) be the region bounded above by the curve

y=\dfrac{kx}{x+3}

, below by the

x

-axis, and on the right by the vertical line

x=k

. The region R(k) is revolved about the

x

-axis to generate a solid S(k). (a) Express the volume

V(k)

of S(k) as a definite integral in terms of

k

and evaluate the integral to obtain

V(k)

as an explicit function of

k

. (b) Determine all positive values of

k

for which

V(k)=18\pi

. Justify that any solutions you obtain satisfy

k>0

. (c) Find the value of

k

that minimizes the volume

V(k)

, and verify that your result corresponds to a minimum.