Let be the region in the first quadrant bounded above by the curve and below by the…

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

Problem

Let

R

be the region in the first quadrant bounded above by the curve

y=\sqrt{x}

and below by the

x

–axis for

0\le x\le 4

. The region is revolved about the vertical line

x=a

, where

0< a<4

, producing a solid whose volume depends on the parameter

a

. (a) Derive a closed–form expression

V(a)

for the volume of the solid in terms of

a

. Show all steps, including the determination of appropriate limits of integration. (b) Determine the value of

a

in the open interval

(0,4)

that minimizes

V(a)

. Use calculus to justify that your result gives a minimum, not a maximum. (c) Interpret geometrically why the value of

a

found in part (b) yields the smallest possible volume for this family of solids.