Mathematics

Problem

Consider the function

y=\sqrt{x}

on the interval

0\le x\le 4

. The region bounded by the graph of

y=\sqrt{x}

, the

x

-axis, and the line

x=4

is revolved about the

x

-axis to generate a solid. (a) Derive an explicit integral expression for the exact volume

V

of the solid. (b) The graph is vertically stretched so that every

y

-value is multiplied by a positive constant

c

. Write an integral expression for the corresponding volume

V(c)

, and show that

\dfrac{V(c)}{V}=c^{2}

. (c) Evaluate

\dfrac{V(c)}{V}

when

c=2

, and interpret the result in terms of how the stretching factor affects the volume.