A region W in the plane is bounded above by the graph of

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

Problem

y=4-x^{2}

, bounded below by the

x

-axis, and bounded on the left by the

y

-axis. The region is revolved about the horizontal line

y=d

, where

d>4

, producing a solid whose volume depends on

d

. (a) Derive, in terms of

d

, an explicit formula

V(d)

for the volume of the solid. Express your answer as a single definite integral evaluated to a closed-form expression containing

d

. (b) Use your result from part (a) to determine

V(6)

, the volume when the region is revolved about

y=6

. Provide the exact value. (c) Determine the limit of

V(d)

d\to\infty

, and explain geometrically why this limit is reasonable.