Consider the region that lies below the curve

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

Problem

Consider the region

V

that lies below the curve

y=4-x^{2}

, above the

x

–axis, and between

x=-2

and

x=2

. For a real number

k

with

k>4

, the region is revolved about the horizontal line

y=k

, producing a solid whose volume depends on

k

. (1) Derive an explicit formula

V(k)

for the volume of the solid in terms of

k

. Show all steps leading from the geometric description of the solid to the final simplified algebraic expression. (2) For

4<k<10

, determine the value of

k

for which

V(k)=150\pi

. Express your answer exactly, then provide a decimal approximation correct to three decimal places. (3) Compute

\displaystyle \frac{dV}{dk}

and interpret its numerical value in the context of this problem. Explain, in a sentence, what this rate of change says about how the volume responds to incremental increases in

k

. (No units beyond those implied in the problem are required.)

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