Mathematics

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

. a. 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. b. 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. c. 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.)

Intuitive explanation

When you revolve a region around a horizontal line that's above the region, you're creating a washer-shaped solid where the outer radius is larger than the inner radius. The key insight is that both radii depend on the distance from the axis of rotation $y=k$ to the parabola and to the $x$ -axis—and these distances change as $k$ varies. Set up the washer method carefully: the outer radius is $k$ minus the lowest point, and the inner radius is $k$ minus the parabola's height, then integrate to find how the volume formula depends on $k$ .

Intuitive explanation

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